Hello (again) SeqFan & Math-fun,
Here are a few Kimberlike sequences computed by Gilles Sadowski
(for k=2 to k=10 ; see a short explanation here),
Best,
É.
[k=2 is
A026272]
(k=2): 1,2,1,3,2,4,5,3,6,7,4,8,5,9,10,6,11,7,12,13,8,14,15,9,16,10,17,18,11,19,20,12,21,13,22,23,14,24,15,25,26,16,27,28,17,29,18,30,31,19,32,20,33,34,21,35,36,22,37,23,38,39,24,40,41,25,42,26,43,44,27,45,28,46,47,29,48,49,30,50,31,51,52,32,53,54,33,55,34,56,57,35,58,36,59,60,37,61,62,38,63,39,64,65,40,66,41,67,68,42,69,70,43,71,44,72,73,45,74,75,46,76,47,77,78,48,79,49,80,81,50,82,83,51,84,52,85,86,53,87,54,88,89,55,90,91,56,92,57,93,94,58,95,96,59,97,60,98,99,61,100,62,101,102,63,103,104,64,105,65,106,107,66,108,109,67,110,68,111,112,69,113,70,114,115,71,116,117,72,118,73,119,120,74,121,75,122,123,76,124,125,77,126,78,127,128,79,129,130,80,131,81,132,133,82,134,83,135,136,84,137,138,85,139,86,140,141,87,142,143,88,144,89,145,146,90,147,91,148,149,92,150,151,93,152,94,153,154,95,155,96,156,157,97,158,159,98,160,99,161,162,100,163,164,101,165,102,166,167,103,168,104,169,170,105,171,172,106,173,107,174,175,108,176,109,177,178,110,179,180,111,181,112,182,183,113,184,185,114,186,115,187,188,116,189,117,190,191,118,192,193,119,194,120,195,196,121,197,198,122,199,123,200,201,124,202,125,203,204,126,205,206,127,207,128,208,209,129,210,130,211,212,131,213,214,132,215,133,216,217,134,218,219,135,220,136,221,222,137,223,138,224,225,139,226,227,140,228,141,229,230,142,231,143,232,233,144,234,235,145,236,146,237,238,147,239,240,148,241,149,242,243,150,244,151,245,246,152,247,248,153,249,154,250,251,155,252,253,156,254,157,255,256,158,257,159,258,259,160,260,261,161,262,162,263,264,163,265,164,266,267,165,268,269,166,270,167,271,272,168,273,274,169,275,170,276,277,171,278,172,279,280,173,281,282,174,283,175,284,285,176,286,287,177,288,178,289,290,179,291,180,292,293,181,294,295,182,296,183,297,298,184,299,185,300,301,186,302,303,187,304,188,305,306,189,307,308,190,309,191,310,311,192,312,193,313,314,194,315,316,195,317,196,318,319,197,320,198,321,322,199,323,324,200,325,201,326,327,202,328,329,203,330,204,331,332,205,333,206,334,335,207,336,337,208,338,209,339,340,210,341,342,211,343,212,344,345,213,346,214,347,348,215,349,350,216,351,217,352,353,218,354,219,355,356,220,357,358,221,359,222,360,361,223,362,363,224,364,225,365,366,226,367,227,368,369,228,370,371,229,372,230,373,374,231,375,376,232,377,233,378,379,234,380,235,381,382,236,383,384,237,385,238,386,387,239,388,240,389,390,241,391,392,242,393,243,394,395,244,396,397,245,398,246,399,400,247,401,248,402,403,249,404,405,250,406,251,407,408,252,409,253,410,411,254,412,413,255,414,256,415,416,257,417,418,258,419,259,420,421,260,422,261,423,424,262,425,426,263,427,264,428,429,265,430,431,266,432,267,433,434,268,435,269,436,437,270,438,439,271,440,272,441,442,273,443,274,444,445,275,446,447,276,448,277,449,450,278,451,452,279,453,280,454,455,281,456,282,457,458,283,459,460,284,461,285,462,463,286,464,287,465,466,288,467,468,289,469,290,470,471,291,472,473,292,474,293,475,476,294,477,295,478,479,296,480,481,297,482,298,483,484,299,485,486,300,487,301,488,489,302,490,303,491,492,304,493,494,305,495,306,496,497,307,498,308,499,500,309,501,502,310,503,311,504,505,312,506,507,313,508,314,509,510,315,511,316,512,513,317,514,515,318,516,319,517,518,320,519,520,321,521,322,522,523,323,524,324,525,526,325,527,528,326,529,327,530,531,328,532,329,533,534,330,535,536,331,537,332,538,539,333,540,541,334,542,335,543,544,336,545,337,546,547,338,548,549,339,550,340,551,552,341,553,342,554,555,343,556,557,344,558,345,559,560,346,561,562,347,563,348,564,565,349,566,350,567,568,351,569,570,352,571,353,572,573,354,574,575,355,576,356,577,578,357,579,358,580,581,359,582,583,360,584,361,585,586,362,587,363,588,589,364,590,591,365,592,366,593,594,367,595,596,368,597,369,598,599,370,600,371,601,602,372,603,604,373,605,374,606,607,375,608,376,609,610,377,611,612,378,613,379,614,615,380,616,617,381,618,382,...
(k=3): 1,3,1,4,1,3,5,6,4,3,8,10,5,4,6,7,9,11,5,8,13,6,10,7,2,12,9,2,8,11,2,7,14,10,13,16,9,17,12,18,15,11,20,21,22,24,19,14,13,25,23,12,16,26,27,17,15,30,18,31,28,32,14,20,34,21,19,22,33,16,24,36,15,17,23,25,35,18,37,29,26,38,27,40,20,42,19,21,30,28,22,31,39,41,32,24,43,44,23,34,45,25,33,46,47,49,50,26,36,29,27,48,35,52,53,54,37,51,28,30,38,56,57,31,40,55,58,32,42,59,60,61,39,62,34,41,33,63,64,29,43,65,44,66,67,36,45,68,35,70,46,71,47,72,37,49,73,50,69,38,48,74,76,77,78,40,52,79,53,51,54,42,39,80,75,81,83,41,56,82,57,55,84,85,43,58,87,44,89,59,90,60,45,61,91,86,62,46,88,92,47,63,93,64,95,49,96,65,50,48,66,94,67,98,99,101,68,97,102,52,70,51,53,71,100,54,72,103,69,104,73,105,107,108,111,56,74,55,57,76,106,77,110,78,58,112,109,79,114,59,75,113,60,116,80,61,117,81,118,62,83,119,82,115,121,63,122,84,64,85,123,120,124,65,87,126,127,66,89,125,67,90,86,128,130,68,91,88,129,131,132,70,92,133,134,71,93,135,69,72,95,136,137,96,73,138,94,139,140,142,144,74,98,143,99,97,76,101,141,77,146,102,78,145,147,100,75,79,148,150,151,103,152,149,104,80,153,105,154,81,107,155,108,156,83,82,111,106,157,159,160,162,84,110,161,85,109,163,112,164,165,158,87,114,168,113,169,166,89,86,116,167,90,170,117,171,88,118,91,115,173,119,174,175,177,92,121,172,176,122,93,178,120,179,123,180,95,124,181,182,96,94,126,184,127,125,185,183,187,188,192,98,128,97,99,130,190,186,129,101,189,131,193,132,102,191,100,133,194,134,196,197,199,135,195,103,198,201,136,104,137,200,202,105,138,203,204,139,107,140,205,108,142,206,106,144,208,143,111,207,141,209,210,211,110,214,109,146,212,145,213,112,147,216,218,219,220,148,114,113,150,217,151,149,215,152,221,116,222,223,153,224,117,154,225,115,118,155,226,227,156,119,228,230,231,232,157,229,121,159,233,160,122,120,162,234,161,236,123,238,158,163,124,235,164,237,165,239,240,126,241,125,127,168,166,243,169,244,242,167,245,128,246,248,170,249,130,171,129,251,252,247,253,131,173,255,132,174,254,175,172,133,177,250,134,176,256,257,259,135,178,258,260,179,261,136,180,262,137,264,181,265,182,138,266,268,263,139,184,267,140,183,185,269,270,142,187,271,188,272,144,143,141,192,186,273,275,190,276,278,189,274,277,279,146,145,193,191,280,281,147,282,283,194,284,285,148,196,290,197,195,150,199,149,151,198,287,288,152,201,286,289,200,291,153,202,292,294,154,203,293,204,295,155,296,297,205,156,298,299,206,300,301,302,157,208,303,207,304,159,305,209,160,210,306,211,307,162,161,158,214,212,309,310,213,163,311,312,313,164,216,314,165,218,308,219,315,220,316,217,215,166,168,317,320,322,169,167,221,319,323,222,318,223,324,170,224,325,321,171,225,326,327,328,329,226,330,227,173,331,332,228,174,172,230,175,231,229,232,333,177,334,176,335,233,336,337,338,341,178,234,339,340,179,236,343,344,180,238,235,345,342,181,237,346,182,348,239,350,240,351,349,241,184,352,183,347,353,185,243,354,242,244,355,187,356,245,188,357,246,358,186,248,360,361,249,192,359,190,247,189,251,362,252,364,365,253,366,363,191,193,255,368,254,250,369,367,370,194,371,372,373,375,256,196,257,195,197,259,374,258,376,199,260,198,377,261,378,379,380,262,201,381,200,264,382,383,265,202,384,263,266,386,203,268,385,204,267,387,389,391,392,205,269,388,270,390,206,393,271,394,395,272,396,208,207,398,399,400,273,402,209,275,397,210,276,274,211,278,401,277,403,404,279,212,214,406,408,213,280,407,281,411,413,282,405,283,216,409,284,412,285,218,410,414,219,215,217,220,290,415,416,417,418,287,419,288,286,423,424,221,289,420,421,222,291,422,223,426,292,427,224,294,428,293,425,225,429,295,430,431,296,226,297,432,227,433,298,435,299,228,436,300,437,301,230,302,229,231,438,303,232,439,304,434,440,305,442,233,443,445,306,441,444,307,234,446,447,448,449,450,236,309,451,310,235,452,238,311,453,312,237,313,308
(k=4): 1,3,1,6,1,3,1,7,5,3,6,10,8,3,5,7,11,6,13,14,5,8,10,7,6,9,5,15,11,17,8,7,13,10,14,9,16,12,19,8,11,24,18,15,10,9,13,17,2,14,12,2,11,16,2,9,22,2,19,15,13,18,4,12,14,17,24,4,20,23,16,25,4,26,21,15,12,4,19,22,18,27,29,17,30,33,34,16,31,20,35,24,36,23,37,32,21,25,19,18,26,28,22,39,40,41,44,42,46,27,20,48,29,43,49,30,24,23,21,33,31,34,50,25,51,22,35,26,32,36,28,20,37,45,47,38,53,27,58,55,21,23,29,39,54,40,30,41,56,25,42,44,31,33,26,46,34,43,57,28,48,32,35,62,49,27,36,52,60,63,37,65,29,50,38,59,51,30,61,45,66,64,47,39,31,67,40,33,28,41,53,34,68,42,32,55,44,58,35,54,69,43,46,36,71,56,74,75,37,48,73,70,78,38,49,80,57,76,81,72,52,82,83,39,50,45,62,40,51,60,47,41,77,63,85,59,42,65,79,84,61,44,86,87,53,43,64,66,88,46,91,55,38,67,54,90,58,92,48,89,96,68,56,98,49,94,100,95,103,104,69,45,93,52,57,50,71,97,47,105,51,74,70,75,73,106,99,108,111,62,60,78,72,101,76,59,80,63,53,112,81,109,61,65,82,115,83,55,116,54,77,64,102,113,66,58,119,110,79,56,85,67,107,117,84,118,52,114,121,86,68,87,57,122,127,129,120,88,124,131,69,125,91,128,132,134,90,135,71,89,92,60,62,70,130,59,74,96,73,75,94,63,98,95,61,72,93,100,126,65,78,76,103,136,104,97,64,80,137,140,123,66,81,133,138,105,99,82,77,67,83,139,106,146,148,101,108,149,79,68,111,143,141,144,147,152,85,153,155,84,69,109,112,154,150,102,86,156,151,87,71,115,158,142,70,116,88,113,110,157,107,74,73,162,91,75,119,90,72,89,159,117,114,92,118,145,160,161,76,78,121,165,167,169,96,94,122,120,80,95,93,98,127,124,81,129,77,125,100,163,82,131,128,97,83,164,103,132,79,104,134,166,173,135,175,130,99,170,168,174,176,105,85,171,84,126,172,101,179,106,180,177,181,86,123,108,188,87,192,193,136,184,111,190,191,88,137,133,102,178,140,109,189,138,183,112,91,186,90,89,196,139,199,187,182,197,92,115,107,110,146,113,116,148,141,143,194,149,144,198,94,96,147,185,93,95,119,114,152,117,98,153,195,118,155,150,142,201,154,100,151,97,121,200,156,202,203,204,120,122,103,158,206,207,104,208,99,157,124,210,127,145,211,125,212,129,162,105,213,101,159,128,214,131,209,215,106,160,216,161,132,219,222,205,108,134,130,165,221,135,167,102,126,169,111,217,218,223,123,220,224,109,230,163,233,234,235,227,112,231,164,237,239,241,242,243,107,136,166,225,110,133,240,115,137,113,173,168,170,116,175,140,138,244,174,171,229,176,236,172,245,139,114,232,238,247,119,117,179,177,246,180,118,226,181,248,141,146,250,143,251,252,148,144,121,188,184,149,178,120,147,192,122,193,190,249,191,254,183,142,256,152,189,124,186,153,150,182,127,125,155,187,151,154,228,129,258,196,259,253,128,156,199,197,255,131,257,145,262,185,158,194,263,132,265,157,260,130,198,126,134,261,123,268,272,135,266,275,195,162,159,267,271,264,274,278,279,281,160,201,269,161,280,283,200,284,273,277,202,285,203,287,204,165,136,133,290,282,167,206,291,207,137,169,208,276,270,163,296,298,210,138,140,292,211,164,293,212,297,294,288,139,213,295,209,166,299,214,300,205,289,215,301,304,311,216,168,307,141,170,173,219,305,143,146,222,175,171,144,174,221,148,172,217,176,218,302,149,147,306,220,223,142,286,303,224,308,177,179,309,314,310,180,230,152,227,181,150,233,153,234,231,235,312,151,225,155,178,154,237,315,184,239,145,188,241,317,242,156,243,318,316,183,240,192,190,229,193,191,158,182,186,189,157,313,236,232,319,244,187,226,320,325,238,326,323,245,324,331,329,159,332,162,247,196,185,328,246,322,160,330,197,199,161,248,327,194,334,335,250,336,338,251,341,252,342,347,198,333,165,344,321,348,228,195,249,167,340,337,343,346,163,254,169,339,345,350,256,351,352,164,201,355,356,362,200,365,368,369,349,358,202,166,253,203,258,367,204,259,353,255,366,359,371,257,168,206,372,360,207,170,262...
(k=5): 1,3,1,7,1,3,1,8,1,3,9,7,2,3,13,2,8,3,2,7,9,2,10,12,2,8,18,7,13,11,9,14,19,10,8,7,12,16,20,23,9,11,13,8,10,18,14,25,29,12,9,22,19,11,16,10,13,24,27,20,28,14,12,23,18,11,10,17,37,26,13,16,19,25,22,12,14,11,29,30,20,33,24,18,31,17,27,23,16,28,35,14,19,36,32,41,26,22,46,25,52,20,18,17,47,16,37,24,29,44,30,23,19,45,27,33,31,48,28,38,22,17,20,26,49,25,35,32,42,51,36,55,24,39,43,23,54,41,29,17,34,30,27,22,37,46,60,28,31,33,26,25,47,52,44,40,59,24,38,45,32,57,35,61,21,62,48,36,29,68,27,42,30,39,49,34,28,26,43,41,31,51,37,33,64,69,21,55,58,70,63,54,46,32,67,74,40,38,35,44,47,56,78,30,36,45,52,60,21,73,34,80,31,39,42,48,59,33,50,57,37,41,43,79,49,61,32,88,62,65,21,72,75,51,35,66,38,40,68,46,76,36,84,55,44,34,54,58,47,64,77,45,21,39,63,69,85,42,56,52,70,81,67,41,48,87,43,82,60,50,74,90,53,92,49,38,59,57,40,83,34,78,95,73,89,51,46,61,86,44,94,62,80,39,97,65,47,45,98,55,42,54,66,79,72,101,58,68,75,103,43,104,52,48,64,56,88,76,63,40,50,96,91,108,49,69,53,84,77,60,67,70,93,106,5,57,59,51,99,102,5,107,85,81,110,74,5,119,112,61,82,105,5,87,62,55,54,73,5,109,78,65,90,83,113,58,92,111,116,66,118,50,56,80,89,86,68,72,95,64,53,120,63,79,75,94,117,15,121,129,60,130,97,57,76,69,59,98,67,100,125,122,70,15,128,88,77,101,131,71,114,61,84,103,91,123,104,62,96,15,74,124,50,127,58,81,93,65,85,56,133,73,108,82,53,15,66,138,99,78,106,87,102,142,64,68,63,83,132,107,72,15,137,90,80,110,146,105,148,92,75,112,86,79,89,69,67,119,115,109,126,76,4,70,95,140,143,4,113,111,94,71,4,134,77,116,147,4,53,118,97,135,4,65,88,74,98,84,156,141,100,149,120,66,117,73,91,81,150,101,121,152,144,153,85,96,82,103,78,129,93,104,130,72,122,114,125,145,139,87,151,83,155,128,157,80,75,158,99,123,131,108,162,79,90,102,124,106,76,86,161,127,92,71,89,160,136,107,163,170,159,164,77,105,133,173,110,167,154,176,95,177,168,178,112,94,138,180,181,109,184,132,84,88,115,183,142,119,97,81,137,111,113,174,172,98,165,126,91,82,85,100,116,146,169,175,182,148,118,185,186,101,96,179,93,83,140,87,134,189,143,103,117,120,188,71,104,135,193,195,197,205,121,192,147,90,86,204,99,198,114,141,190,191,89,92,171,122,102,129,108,149,125,130,106,156,200,144,201,150,202,194,128,123,152,107,95,153,139,105,94,124,131,145,196,199,166,110,203,127,208,187,151,206,210,214,97,112,155,109,91,224,157,136,98,211,158,71,133,219,115,212,100,111,213,162,113,119,93,96,218,225,161,101,132,138,160,215,207,116,159,223,163,154,126,103,164,118,137,142,170,104,221,222,217,167,226,233,99,173,117,168,231,232,120,209,176,229,234,177,146,102,178,134,121,114,148,140,180,108,181,106,165,135,143,184,220,172,174,183,122,237,230,107,169,105,236,216,125,129,244,175,147,141,130,242,243,123,110,182,238,128,239,179,246,185,124,186,247,109,112,149,144,100,131,228,252,127,139,189,150,241,156,188,249,111,115,152,171,145,113,153,193,245,235,195,192,119,197,240,136,248,133,190,151,191,116,205,198,250,254,251,204,166,155,257,118,132,253,227,157,126,264,268,138,158,194,200,117,256,201,259,273,202,263,120,137,275,162,187,114,196,142,266,161,199,121,160,154,267,159,260,203,262,265,272,163,269,134,208,206,164,271,122,277,210,278,146,140,135,214,170,258,167,125,148,261,211,143,129,168,123,280,173,212,130,224,274,213,219,128,124,255,176,279,141,207,177,165,218,147,178,115,215,281,127,131,225,290,180,172,285,181,174,169,223,276,144,139,287,184,149,217,183,221,209,222,175,283,270,292,150,288,226,289,145,133,136,293,156,152,233,182,179,231,153,232,229,126,296,132,185,298,234,186,301,151,220,302,308,171,300,286,138,216,303,305,282,309,189,155,...
(k=6): 1,3,1,7,1,3,1,10,1,3,1,7,11,3,5,12,14,3,10,7,5,3,16,23,11,19,5,7,12,10,21,14,5,24,15,7,11,25,5,16,10,12,34,7,5,19,14,23,11,29,15,10,21,33,12,28,16,35,24,18,11,14,10,25,27,19,15,12,31,40,51,23,11,16,21,48,14,34,18,29,12,49,15,24,28,19,39,33,22,25,16,14,27,35,41,23,21,18,15,37,31,42,53,45,54,19,43,16,24,29,40,22,34,28,15,25,18,44,21,23,27,33,51,55,48,19,39,57,59,35,62,49,31,24,22,18,41,37,66,29,21,25,28,23,42,65,61,34,27,45,43,40,56,64,18,33,53,22,24,54,63,74,44,6,31,35,39,25,70,29,6,28,80,48,51,37,27,6,41,55,22,49,34,50,6,57,67,42,59,33,38,6,40,62,43,45,31,69,6,29,28,35,73,22,27,66,39,44,61,56,53,65,46,37,54,77,78,34,64,60,41,72,48,33,63,86,51,97,31,38,42,49,82,40,50,55,74,35,43,70,4,45,84,57,47,4,39,36,59,85,4,37,44,80,67,4,62,96,58,46,4,89,41,81,53,4,56,69,38,54,61,48,66,42,40,26,73,65,51,93,60,49,43,64,36,50,39,45,63,37,71,55,47,77,72,78,87,44,94,105,95,57,26,100,41,119,46,38,59,79,70,74,86,112,99,82,42,58,53,62,48,36,67,56,54,97,43,84,90,26,51,49,61,45,80,85,50,69,109,66,47,60,44,65,52,81,38,55,64,104,73,89,63,46,96,114,26,88,36,57,103,106,71,101,117,110,75,72,59,121,91,77,53,93,78,123,58,124,116,54,56,70,62,26,87,38,74,50,47,79,67,98,107,94,61,36,95,52,82,86,46,83,60,55,100,105,66,69,111,65,26,113,84,64,99,80,63,57,118,90,129,85,139,97,73,119,112,81,59,122,140,102,36,108,71,58,47,56,50,125,72,89,75,135,131,62,88,46,109,77,52,96,70,78,104,141,61,134,67,130,143,74,91,60,103,101,132,93,106,79,114,144,87,66,76,65,110,69,64,155,63,82,117,146,47,83,86,13,94,136,98,121,95,142,58,116,80,84,73,123,107,13,124,52,126,100,71,85,153,81,90,105,120,72,99,13,157,163,75,138,111,97,148,70,60,113,67,77,160,13,128,89,78,151,102,88,74,118,154,112,158,159,108,13,161,119,175,69,96,79,129,76,122,58,91,127,52,13,109,104,87,93,139,133,82,125,152,101,103,83,92,140,73,86,170,106,131,80,71,135,176,184,84,94,145,114,72,110,95,98,130,81,134,85,75,166,177,141,182,132,117,90,167,168,143,77,100,191,107,52,181,78,116,121,99,188,144,105,186,97,192,89,171,123,88,164,136,124,76,79,146,126,111,120,172,155,142,102,173,113,193,199,150,194,190,96,91,82,87,178,71,108,112,83,189,93,118,197,138,128,86,217,153,92,104,119,187,84,109,101,75,148,103,81,157,122,205,147,94,85,129,163,106,127,95,151,196,209,98,160,156,125,90,154,195,68,207,133,110,158,114,159,139,210,214,76,161,100,131,79,180,89,140,88,99,135,107,97,117,152,202,130,175,105,162,200,185,134,216,116,145,179,132,169,215,121,87,83,91,141,102,206,96,149,111,170,75,123,143,93,224,208,124,113,120,201,92,176,126,198,108,144,166,136,68,112,184,231,115,104,167,101,168,94,177,118,103,146,109,142,95,182,76,90,220,119,223,98,128,106,164,155,181,150,171,219,204,138,122,191,226,212,88,228,188,186,232,172,100,110,129,127,173,192,99,114,153,125,230,227,148,83,222,233,107,147,229,235,178,105,91,133,157,68,193,190,117,151,194,239,131,199,139,102,189,163,116,156,154,92,160,135,130,140,187,197,111,121,158,76,221,159,134,237,225,132,241,113,161,108,213,123,152,101,104,120,145,124,112,217,103,141,205,196,115,126,98,162,109,203,195,180,143,149,118,17,106,209,175,240,136,211,68,169,207,119,144,234,183,218,243,179,185,17,210,255,170,128,271,214,110,122,142,202,146,244,200,266,238,166,114,17,107,251,176,253,138,246,167,127,129,168,92,125,150,216,102,155,206,17,215,184,177,249,117,164,268,201,198,242,208,261,174,116,182,133,171,17,111,148,153,68,131,147,181,224,259,264,108,121,172,113,139,130,262,17,173,135,267,151,157,191,186,188,140,123,120,134,132,231,115,204,124,154,156,220,192,178,...
(k=7): 1,3,1,7,1,3,1,12,1,3,1,7,1,3,9,14,19,3,21,7,12,3,29,15,9,3,20,7,32,23,14,18,16,12,9,7,19,41,34,15,21,49,37,7,9,14,12,20,44,16,18,7,29,23,9,15,19,39,36,12,14,32,21,47,9,33,16,56,20,18,52,15,12,34,9,14,19,23,43,41,37,58,29,16,21,12,64,15,18,20,14,49,55,44,32,36,19,39,48,33,16,23,54,15,57,14,21,18,34,60,20,47,29,61,24,72,19,16,37,15,87,41,43,52,56,23,18,32,21,67,51,20,36,33,16,73,19,39,44,24,58,49,29,34,26,18,71,48,55,23,21,64,20,30,66,74,37,54,85,47,32,65,57,41,24,70,43,33,90,36,60,26,29,23,75,61,52,39,34,69,81,56,51,44,30,98,88,63,72,24,110,49,83,32,37,89,48,67,26,58,86,33,29,105,55,41,36,47,87,73,43,62,54,34,24,30,64,39,71,80,57,66,40,96,102,26,32,65,44,52,74,60,37,76,51,33,70,61,56,24,84,49,106,36,85,48,30,41,34,69,75,63,26,109,43,47,101,39,58,90,55,72,81,40,24,67,115,54,124,33,37,97,92,44,62,88,83,30,57,26,36,64,52,73,98,89,51,86,66,41,71,49,60,65,48,56,87,39,43,61,80,110,93,47,40,74,26,70,30,105,76,94,125,123,114,63,55,58,44,69,96,82,54,100,145,84,75,102,68,127,85,53,112,67,72,52,57,62,51,30,99,49,43,48,81,40,64,133,130,106,90,47,56,60,95,66,135,73,101,65,83,61,71,109,88,92,132,79,119,97,140,128,55,141,86,89,58,54,70,63,74,80,115,98,87,53,40,76,52,69,51,91,48,124,57,126,93,68,151,134,62,67,75,148,82,103,94,72,129,56,84,64,110,158,60,105,85,96,120,147,138,160,66,61,100,65,81,40,55,114,102,73,54,161,71,58,131,123,125,53,99,90,51,63,83,112,122,79,117,70,95,167,57,88,139,74,106,127,92,69,101,86,80,62,76,89,68,97,40,67,145,149,87,109,162,60,64,155,75,130,72,133,98,91,154,61,163,82,119,66,93,65,135,53,143,84,157,178,115,132,128,94,85,121,136,73,71,63,81,103,183,140,169,175,96,141,171,165,124,77,70,105,126,110,156,100,62,79,83,69,74,184,90,102,134,68,142,67,129,120,99,76,88,80,114,153,95,53,86,198,151,92,148,72,75,89,66,65,112,138,87,123,101,106,125,97,117,147,131,122,82,63,158,150,185,91,179,166,71,73,84,98,160,109,93,127,168,118,85,62,81,70,180,181,139,161,94,77,191,69,182,53,177,187,68,74,190,79,119,116,130,96,83,103,115,133,145,152,76,167,189,149,194,90,80,121,100,105,135,128,75,88,132,155,110,86,102,162,143,154,99,136,95,124,92,89,197,144,87,126,163,172,140,82,157,53,170,141,217,114,120,173,204,97,101,84,206,91,129,134,106,112,81,68,85,77,178,186,93,142,98,156,169,165,117,123,146,79,171,175,125,94,122,109,183,76,83,193,138,153,131,148,151,210,222,80,118,214,96,238,224,195,203,127,90,147,184,103,233,159,88,188,86,208,150,235,116,100,119,164,115,158,223,105,139,89,82,92,95,102,99,130,68,160,166,110,225,121,198,133,84,219,221,77,237,168,179,161,128,91,185,97,145,260,135,132,101,79,124,215,152,149,93,180,114,181,126,106,136,98,177,120,143,182,112,216,94,167,230,155,191,187,144,140,154,229,236,190,129,141,162,218,117,109,134,96,228,232,212,123,189,241,157,234,163,122,125,194,231,82,239,103,142,220,172,118,240,170,100,259,92,77,131,146,95,173,113,138,156,105,99,127,102,116,197,200,115,79,119,165,148,91,169,153,151,178,110,251,196,97,171,245,255,147,186,175,93,204,101,121,213,261,130,206,139,150,217,265,183,159,282,94,227,133,106,252,286,128,304,114,158,193,205,124,164,266,288,112,132,137,135,184,120,126,160,296,313,195,145,77,188,166,263,109,210,136,203,285,276,161,117,149,214,152,168,103,143,222,129,100,95,111,208,123,140,224,179,144,122,134,99,141,125,91,118,155,185,198,238,154,113,301,233,258,180,167,223,181,177,246,116,235,115,162,131,142,182,268,119,127,157,219,225,274,221,287,138,280,163,187,146,250,191,226,275,215,254,77,190,271,293,121,237,299,176,156,279,172,170,148,189,...
(k=8): 1,3,1,7,1,3,1,14,1,3,1,7,1,3,1,15,9,3,19,7,20,3,14,30,23,3,9,7,31,3,24,15,28,39,33,7,9,14,19,25,43,20,58,7,44,41,9,15,23,21,18,7,14,40,30,24,9,74,19,7,31,28,20,15,66,25,9,14,33,18,62,21,23,39,75,76,9,79,19,15,24,59,14,20,43,30,9,41,18,44,28,25,31,21,40,15,23,14,19,63,53,58,33,47,20,24,55,18,52,80,86,15,14,39,69,21,30,25,19,28,23,67,71,99,31,20,18,15,43,41,24,66,74,62,44,40,33,21,19,72,46,59,68,25,23,18,20,30,28,82,75,47,76,39,53,24,31,79,19,21,58,52,55,63,18,84,89,20,23,25,33,41,43,57,93,118,40,28,30,44,24,21,94,18,69,60,88,46,31,67,80,50,23,39,71,25,62,86,66,47,95,59,61,21,33,24,28,74,53,30,77,68,72,41,52,85,43,40,55,58,31,25,96,99,44,105,75,63,121,76,110,57,82,39,46,28,116,79,33,87,30,91,50,123,45,119,60,47,128,92,84,102,31,122,69,41,89,67,40,62,43,59,53,115,61,66,71,52,93,44,98,80,33,39,55,88,130,94,58,81,68,46,74,125,86,72,103,126,77,57,45,63,111,50,118,47,95,41,90,40,158,85,75,147,43,143,76,60,129,166,134,153,187,39,44,82,53,79,167,96,52,59,62,99,69,67,61,87,46,117,55,105,66,91,149,84,45,58,71,41,40,110,92,47,50,146,89,57,43,68,121,78,80,116,102,63,145,74,72,44,183,81,93,176,88,119,77,123,60,98,53,86,94,52,174,46,122,128,75,115,150,59,45,76,108,62,55,85,61,90,103,47,95,67,69,50,58,79,82,66,154,137,173,177,111,57,139,130,194,125,71,140,156,118,126,87,96,178,68,63,84,179,46,99,53,91,52,142,45,60,78,72,74,80,203,92,89,105,129,47,212,59,55,81,77,134,50,117,62,143,61,147,110,88,75,58,93,102,86,158,76,67,164,57,69,153,66,94,98,46,116,100,121,85,45,107,166,79,53,52,90,82,71,63,149,119,167,68,95,170,60,123,146,108,103,115,175,50,55,122,187,59,145,87,72,84,128,74,49,78,207,62,61,96,80,155,111,91,180,120,45,57,77,99,131,81,89,193,92,67,75,150,118,66,69,125,197,76,130,137,176,126,88,105,183,86,139,63,50,174,93,60,140,141,71,154,68,79,49,85,102,138,94,110,82,117,45,98,129,156,142,90,100,188,61,160,173,72,200,57,134,177,74,107,95,205,148,116,78,190,84,87,161,143,121,80,178,67,103,147,77,179,108,194,69,119,75,81,49,91,96,115,60,204,76,123,89,195,158,153,191,92,122,99,186,68,71,164,111,114,159,88,86,196,149,128,61,79,203,146,120,70,135,166,93,85,189,82,145,105,72,118,220,212,167,201,131,94,49,226,170,125,90,102,181,239,98,78,126,130,169,155,175,100,110,84,80,137,77,87,95,210,150,117,243,216,139,253,187,81,261,107,198,140,182,141,214,219,129,180,138,91,103,241,116,238,89,96,49,233,154,108,70,142,92,134,121,176,207,86,88,193,144,99,174,119,156,115,183,258,82,85,157,252,197,148,160,143,93,123,111,274,206,211,114,122,78,147,221,231,173,240,94,90,161,105,77,224,49,177,84,120,188,254,128,98,102,87,199,235,118,81,228,153,100,178,95,158,135,200,190,179,249,70,149,110,146,125,131,278,159,205,164,91,145,126,89,267,194,107,130,117,227,230,96,245,103,92,49,88,166,186,191,195,268,137,236,85,204,215,108,116,167,99,318,78,139,155,129,165,196,170,93,150,140,234,189,141,121,138,169,263,203,115,90,119,181,94,175,237,134,260,111,81,70,87,142,201,49,114,133,105,289,123,98,154,212,144,122,102,282,246,95,220,284,100,187,182,295,247,180,143,120,156,148,91,226,283,257,118,157,198,210,128,176,147,110,160,250,174,92,96,216,225,78,239,301,107,255,244,103,214,183,125,135,193,161,219,302,117,131,276,126,287,173,70,207,153,243,108,259,130,197,146,149,90,158,177,116,269,145,253,328,233,238,241,319,290,311,206,159,188,261,137,178,211,320,164,129,115,121,179,306,98,111,119,139,105,199,256,221,190,102,293,114,140,100,166,138,141,200,231,224,155,150,134,252,123,258,294,167,122,240,194,133,165,70,205,186,142,...
(k=9): 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,16,1,3,9,7,10,3,23,15,12,3,38,7,9,3,26,10,16,3,39,7,43,12,9,15,32,19,10,7,40,45,23,61,9,16,12,7,46,10,50,15,53,26,9,7,69,19,66,12,10,38,16,7,9,57,23,15,52,32,39,10,12,62,9,68,43,19,74,16,26,40,10,15,9,12,56,45,80,42,23,81,90,10,9,46,16,19,12,15,38,50,32,87,10,61,53,26,67,41,39,12,109,16,23,15,88,19,76,89,43,52,40,57,12,66,69,78,95,72,16,15,42,45,26,32,62,19,23,38,106,98,46,56,68,77,84,16,124,138,39,41,50,74,93,108,101,19,131,151,53,26,23,40,43,110,102,61,32,80,121,134,150,81,52,42,67,19,38,45,132,57,135,90,159,112,23,105,26,46,39,87,66,41,196,76,69,19,144,62,56,32,72,50,40,88,78,113,43,89,23,96,156,68,53,26,157,38,42,104,161,107,109,77,95,45,139,52,74,61,39,84,164,160,32,41,46,143,227,57,98,166,26,181,67,40,203,106,93,120,80,140,43,195,50,81,38,56,101,66,149,42,62,188,108,202,69,32,53,102,39,45,76,124,90,72,110,41,153,87,52,183,68,46,193,78,40,146,138,141,131,61,121,105,88,38,43,57,112,89,32,77,205,74,42,50,134,192,96,189,39,151,67,132,56,204,84,45,135,41,95,113,53,150,104,62,66,40,211,107,46,80,109,52,38,182,69,81,233,98,43,228,93,144,159,163,206,42,72,76,39,68,130,61,106,57,50,229,238,191,101,41,200,45,78,90,139,87,40,156,120,56,102,108,157,222,53,46,74,77,67,143,161,88,43,209,52,110,62,89,42,196,140,66,160,225,263,164,124,105,137,84,248,41,199,96,69,50,166,45,149,112,80,57,121,61,95,224,173,81,68,72,131,243,46,181,76,138,56,104,53,141,153,42,146,113,93,107,98,52,246,134,109,78,172,41,132,236,67,195,203,62,188,74,135,87,90,77,50,249,66,106,101,151,288,183,227,239,202,158,251,57,88,214,150,102,69,61,193,89,216,221,108,130,53,56,84,268,144,68,152,120,52,80,72,283,258,197,110,189,192,81,96,76,159,105,139,230,205,163,254,219,95,232,62,272,67,265,182,312,204,279,78,124,112,143,156,66,74,57,93,320,157,140,104,77,121,98,137,61,211,215,56,87,161,107,69,90,289,113,300,191,109,206,131,160,175,237,68,295,149,88,164,200,101,322,138,72,106,89,228,84,233,141,80,166,134,62,102,132,76,146,298,81,67,212,153,229,196,293,135,108,173,271,66,209,310,238,222,96,277,78,253,74,199,266,325,181,95,110,250,105,120,77,130,151,69,172,234,297,190,225,305,275,93,150,270,68,158,332,307,87,280,144,313,244,98,112,90,104,72,188,224,152,315,195,139,257,124,183,88,248,67,107,203,80,84,76,121,89,263,341,109,113,159,81,101,243,349,143,282,202,193,163,140,106,137,292,74,328,78,102,303,156,236,299,131,246,214,189,157,77,359,192,227,197,96,216,296,326,108,182,291,221,95,138,161,239,304,249,149,134,132,72,205,141,160,105,93,87,110,324,251,334,358,204,135,164,146,90,362,219,175,120,76,98,80,294,230,88,153,84,166,191,104,130,232,81,301,89,112,184,211,288,284,258,268,215,340,78,206,107,331,306,200,254,173,151,101,77,309,262,109,124,354,333,150,113,144,283,121,316,106,265,102,196,139,158,272,181,172,348,389,96,329,237,152,302,212,228,95,279,380,355,199,209,233,87,352,108,367,190,93,143,137,140,374,229,131,337,90,222,387,105,159,395,84,357,88,290,188,110,312,183,98,371,238,163,289,89,156,195,264,318,138,132,134,225,157,104,397,300,320,234,120,295,253,141,203,185,149,250,135,112,384,271,193,224,161,107,101,360,266,241,202,146,130,160,277,189,244,298,109,293,182,382,192,102,322,106,96,113,164,197,153,386,95,270,124,275,248,379,121,214,391,257,93,310,175,166,368,243,388,280,216,422,336,297,108,151,205,236,227,144,263,221,139,150,305,105,204,366,325,98,246,191,184,390,307,344,173,431,110,376,394,356,158,239,313,152,131,219,137,282,104,335,140,143,427,249,315,393,332,172,211,200,429,206,226,230,292,181,101,215,251,120,112,107,132,232,413,134,196,138,159,299,303,96,430,456,102,475,109,539,291,296,106,156,135,141,...
(k=10): 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[Currently missing integers for k=10 after computation of sequence’s
first 5000 terms:
2,4,6,8,9,10,12,13,14,16,17,20,21,22,24,25,26,29,30,32,33,35,36,37,38,40,41,43,45,47,53,54,56,57,58,60,61,62,63,64,65,67,68,72,74,75,76,80,81,82,83,84,91,92,93,95,99,100,101,104,105,107,112,113,114,116,117,121,122,123,125,127,129,135,136,137,140,144,146,150,152,153,156,158,160,162,163,166,168,169,171,172,177,178,179,180,181,182,183,186,192,196,202,204,205,212,216,220,221,223,224,226,234,235,236,237,245,251,253,256,259,260,263,264,268,271,273,277,281,282,285,287,288,289,290,292,297,298,302,303,305,308,309,315,317,318,322,324,326,328,330,332,333,335,337,339,347,353,368,370,371,373,379,381,382,383,387,388,392,396,400,402,404,406,409,417,423,424,425,434,436,438,442,447,449,450,451,458,463,464,466,469,470,472,476,484,490,498,505,508,510,515,518,524,525,527,536,537,542,546,548,549,554,556,557,561,562,574,575,580,582,584,595,602,611,613,616,620,621,623,625,628,630,636,642,644,645,648,667,668,670,672,680,689,691,692,693,698,699,705,714,716,717,721,728,739,742,752,757,767,768,784,795,796,812,816,819,822,827,828,830,834,837,838,839,843,855,858,859,869,890,891,894,899,901,905,908,909,916,917,920,921,923,929,932,944,947,952,960,964,965,970,986,994,998,999,1000,1001,1005,1006,1009,1011,1012,1016,1023,1024,1025,1029,1054,1061,1064,1070,1073,1077,1078,1083,1090,1092,1093,1095,1097,1101,1118,1122,1123,1125,1132,1133,1135,1137,1140,1145,1151,1162,1165,1170,1173,1179,1180,1184,1187,1196,1199,1203,1209,1210,1212,1222,1226,1231,1232,1237,1242,1243,1245,1249,1251,1254,1255,1258,1261,1268,1269,1270,1271,1272,1274,1275,1281,1282,1283,1285,1286,1287,1288,1289,1290,1291,1297,1298,1299,1300,1301,1304,1306,1307,1312,1313,1316,1318,1319,1320,1323,1326,1328,1329,1331,1333,1338,1339,1341,1342,1345,1346,1349,1351,1352,1354,1355,1356,1357,1359,1363,1364,1365,1372,1374,1376,1377,1380,1381,1382,1383,1387,1388,1389,1392,1397,1398,1400,1401,1402,1404,1405,1407,1408,1409,1410,1412,1413,1414,1417,1418,1419,1420,1422,1423,1424,1426,1427,1428,1429,1430,1432,1433,1434,1435,1437,1438,1439,1441,1442,1443,1445,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1460,1461,1462,1463,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1479,1480,1481,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,...]
P.-S.
A short time after this post, Hugo P.
replied to SeqFan (march 10th, 2006):
----------
I have every confidence that every n will
appear in the sequence for any k, but no idea how you might prove
it.
I knocked up a program to find where 2 would
get inserted for various k:
k n used shown passed seq
2 2 1
1
0 1
3 2
11 24 1
23
4 2
18 48 5
60
5 2 5
12 3 36
6 2
346 1384 14
1785
7 2
5725 26336 70
34759
8 2
9833 50715 180
69998
9 2
38238 217258 734
311739
10 2 33637
211951 1238 313806
For k=3, these mean that when
the first 2 appeared there were 24 numbers to its left in the sequence ('shown'), and 23 numbers (or holes) already known to its right ('seq'); 11 different numbers appeared before the first 2 ('used'), and 1 number had been 'passed' (ie out of 1..13, one number has not
yet been used).
For k = 10, n in (1 .. 100), here are the 'used' values:
(n=1..10) 0 33637 1 79609 5 48647 2 37597 133669 87742
(n=11..20) 9 49117 27191 58164 3 83523 91922 4 13
120918
(n=21..30) 67986 98032 6 82594 38993 43316 7 8 104427
42940
(n=31..40) 10 102548 80105 15 104550 52375 58202 96393
22 42760
(n=41..50) 48638 11 66769 19 73314 28 80498 18 26 14
(n=51..60) 36 12 92532 83185 16 26380 81768 38333 20
103525
(n=61..70) 37957 30080 64703 92274 38747 23 98316
75075 31 32
(n=71..80) 34 38622 17 67139 42371 96870 48 24 25
91412
(n=81..90) 91652 73714 46186 81232 27 30 21 58 33 46
(n=91..100) 65989 98462 65864 63 66631 29 64 39 82235
83836
It took about 17 minutes to find them all; last to appear was 9:
k n used shown passed seq
10 9 133669
840571 9207 1257313
and peak values of 'passed'
and 'seq' (within the 100 results captured) appear at the previous
number:
10 20 120918
764934 21958 1268672
Hope this helps,
Hugo
----------
Thanks -- great job, Hugo!
__________
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