Weighing scales and sequences

http://azoujeremie.files.wordpress.com/2009/09/balance-roberval.jpg

Hello SeqFans,

 

Here is a first succession of empty weighing scales:

 

|     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |

+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+

 

We write under each scale its (un)balance:

 

|     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |

+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+

   0        0        0        0        0        0        0        0        0  

 

We fill each scale with two integer weights:

 

|1   2|  |3   5|  |6   9|  |10 15|  |16 22|  |23 32|  |33 43|  |44 59|  |60 76|

+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+

   1        2        3        5        6        9        10       15       16  

 

Ok, you get it, the unbalances’ seq is the seq formed by the successive weights:

 

S = 1,2,3,5,6,9,10,15,16,22,23,32,33,43,44,59,60,76,...

 

 

S is monotonically increasing and not in the OEIS.

 

If we drop the "monotonically increasing" constraint and want the sequence to be a permutation of the Natural numbers (1,2,3,4,5,6,7,...n) we have:

 

T = 1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,29,43,28,50,...

 

The algorithm used here was, as usual, "take the smallest available integer not yet present in T and not leading to a contradiction". T is not in the OEIS either.

 

Building T is smooth -- except for some weights which have to be delayed:

 

T = 1,2,3,5,4,7,6,11,8,12,9,16,13,19, ... is ok

     1   2   3   5    4    7     6

 

T = 1,2,3,5,4,7,6,11,8,12,9,16,10,--, ... is not (10-6 and 10+6 are already in T)

     1   2   3   5    4    7     6

 

Now, what happens _between_ the weighing scales?

 

For S, the scales are always "separated" by weights of 1 unit:

 

S = |1   2|   |3   5|   |6   9|   |10 15|   |16 22|   |23 32|   |33 43|   |44 59|   |60 76|

    +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+

       1    ^    2    ^    3    ^    5    ^    6    ^    9    ^    10   ^    15   ^    16  

 

Could the succession of the separations be the sequence itself? Let’s try:

 

S’ = |1   2|   |3   5|   |7  10|   |13 18|   |23 30|   |37 47|    |57 70|    |83  101|  

     +--.--+ 1 +--.--+ 2 +--.--+ 3 +--.--+ 5 +--.--+ 7 +--.--+ 10 +--.--+ 13 +---.---+  ...

        1    ^    2    ^    3    ^    5    ^    7    ^    10   ^^    13   ^^     18        

 

It works... but S’ is already in the OEIS: http://www.research.att.com/~njas/sequences/A033485

with the definition: "a(n) = a(n-1) + a([n/2]), a(1) = 1"

 

Now the difficult part: could we build a sequence similar to S, but dropping the "monotonically increasing" constraint?

 

We are thus looking for a sequence T’ where:

- a(n) is not always > a(n-1)

- a(n) doesn’t show twice

- the succession of the separations (between successive scales) form T’ itself

 

I think T’ is not impossible to construct and might start like this:

 

T’ = |1   2|   |3   5|   |7  10|   |13 18|   |23 16|   |9  19|    |29 42|    |55 37|  

     +--.--+ 1 +--.--+ 2 +--.--+ 3 +--.--+ 5 +--.--+ 7 +--.--+ 10 +--.--+ 13 +--.--+  ...

        1    ^    2    ^    3    ^    5    ^    7    ^    10   ^^    13   ^^    18        

 

Could T’ be a permutation of the Naturals? Mmmmmh...

 

Best,

É.

 

__________

 

 

[Douglas McNeil, a couple of hours later, June 14th, 2010]:

 

I believe I can confirm your values for the easier sequences S, T, and S’:

 

sage: S[:50]

[1, 2, 3, 5, 6, 9, 10, 15, 16, 22, 23, 32, 33, 43, 44, 59, 60, 76, 77, 99, 100, 123, 124, 156, 157, 190, 191, 234, 235, 279, 280, 339, 340, 400, 401, 477, 478, 555, 556, 655, 656, 756, 757, 880, 881, 1005, 1006, 1162, 1163, 1320]

 

sage: T[:50]

[1, 2, 3, 5, 4, 7, 6, 11, 8, 12, 9, 16, 13, 19, 10, 21, 14, 22, 15, 27, 17, 26, 18, 34, 20, 33, 23, 42, 25, 35, 24, 45, 29, 43, 28, 50, 31, 46, 30, 57, 32, 49, 36, 62, 37, 55, 38, 72, 39, 59]

 

sage: S’[:50]

[1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 119, 142, 165, 195, 225, 262, 299, 346, 393, 450, 507, 577, 647, 730, 813, 914, 1015, 1134, 1253, 1395, 1537, 1702, 1867, 2062, 2257, 2482, 2707, 2969, 3231, 3530, 3829, 4175, 4521, 4914]

 

But I disagree with the last value of T’, 37: I think the pair (55,37) blocks and you have to use 73.

 

sage: T’[:50]

[1, 2, 3, 5, 7, 10, 13, 18, 23, 16, 9, 19, 29, 42, 55, 73, 91, 68, 45, 61, 77, 86, 95, 114, 133, 104, 75, 117, 159, 214, 269, 342, 415, 324, 233, 165, 97, 142, 187, 248, 309, 232, 155, 241, 327, 422, 517, 631, 745, 612]

 

 

Doug

 

__________

 

[me, Eric]:

 

Thank you, Doug!

 

> But I disagree with the last value of T’, 37:

 

... yes, you are right, I was not careful enough, thanks!

 

Best,

É.

 

__________

 

[February 2014 update]

 

I had forgotten about this page in February 2014 – and reposted on SeqFans the same question (about T) -- in other words (shame on me!):

 

Hello SeqFans,

Here is a permutation of the Naturals not yet in the OEIS, I think:

 

T=1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,...

 

Put parentheses around each pair of integers, like this:

 

T=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),(13,19),(10,21),(14,22),(15,27),(17,26),(18,34),(20,33),(23,42),(25,35),(24,45),...

 

If you replace each pair of integers by their difference, you'll get T again:

 

T=1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,...

 

T was always extended with the smallest unused term not leading to a contradiction.

 

For example, after:

 

T=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),...

 

... one cannot extend T with 10 as 10 would produce the pair (10,16) with a "16" being already used.

 

... similarly, after:

 

T=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),(13,19),(10,21),(14,22),(15,27),(17,26),(18,34),(20,33),(23,42),...

 

... one cannot extend T with 24 as 24 would produce the pair (24,34) with a "34" being already used.

 

Best,

É.

__________

 

Here is Jean-Marc Falcoz’ answer to this post, which came together with 3 graphs of T (the 4th and last one comes from Hans Havermann):

 

T = 1, 2, 3, 5, 4, 7, 6, 11, 8, 12, 9, 16, 13, 19, 10, 21, 14, 22, 15, 27, 17, 26, 18, 34, 20, 33, 23, 42, 25, 35, 24, 45, 29, 43, 28, 50, 31, 46, 30, 57, 32, 49, 36, 62, 37, 55, 38, 72, 39, 59, 40, 73, 41, 64, 44, 86, 51, 76, 47, 82, 53, 77, 48, 93, 52, 81, 54, 97, 56, 84, 58, 108, 60, 91, 61, 107, 65, 95, 63, 120, 66, 98, 67, 116, 68, 104, 69, 131, 74, 111, 70, 125, 71, 109, 75, 147, 78, 117, 79, 138, 83, 123, 80, 153, 85, 126, 87, 151, 88, 132, 89, 175, 90, 141, 92, 168, 96, 143, 94, 176, 99, 152, 100, 177, 101, 149, 102, 195, 103, 155, 105, 186, 106, 160, 110, 207, 113, 169, 112, 196, 114, 172, 115, 223, 118, 178, 119, 210, 121, 182, 122, 229, 124, 189, 127, 222, 128, 191, 129, 249, 133, 199, 130, 228, 134, 201, 135, 251, 136, 204, 137, 241, 139, 208, 140, 271, 142, 216, 144, 255, 145, 215, 148, 273, 146, 217, 150, 259, 156, 231, 154, 301, 157, 235, 158, 275, 159, 238, 161, 299, 162, 245, 163, 286, 164, 244, 165, 318, 167, 252, 166, 292, 170, 257, 171, 322, 173, 261, 174, 306, 179, 268, 180, 355, 184, 274, 183, 324, 185, 277, 181, 349, 187, 283, 188, 331, 190, 284, 192, 368, 194, 293, 193, 345, 197, 297, 198, 375, 202, 303, 203, 352, 200, 302, 205, 400, 206, 309, 209, 364, 211, 316, 212, 398, 213, 319, 214, 374, 218, 328, 219, 426, 220, 333, 221, 390, 224, 336, 225, 421, 226, 340, 227, 399, 232, 347, 230, 453, 233, 351, 234, 412, 237, 356, 236, 446, 239, 360, 240, 422, 243, 365, 242, 471, 246, 370, 247, 436, 250, 377, 248, 470, 253, 381, 254, 445, 256, 385, 258, 507, 260, 393, 262, 461, 264, 394, 263, 491, 267, 401, 265, 466, 269, 404, 266, 517, 270, 406, 272, 476, 276, 413, 278, 519, 279, 418, 280, 488, 285, 425, 281, 552, 282, 424, 287, 503, 288, 432, 289, 544, 290, 435, 291, 506, 294, 442, 295, 568, 298, 444, 296, 513, 300, 450, 304, 563, 307, 463, 305, 536, 308, 462, 310, 611, 311, 468, 312, 547, 314, 472, 313, 588, 315, 474, 317, 555, 320, 481, 321, 620, 323, 485, 325, 570, 326, 489, 327, 613, 329, 493, 330, 574, 332, 497, 334, 652, 335, 502, 337, 589, 338, 504, 339, 631, 341, 511, 342, 599, 343, 514, 344, 666, 348, 521, 346, 607, 350, 524, 353, 659, 354, 533, 357, 625, 358, 538, 359, 714, 361, 545, 362, 636, 363, 546, 366, 690, 369, 554, 367, 644, 372, 553, 371, 720, 373, 560, 378, 661, 376, 564, 379, 710, 382, 572, 380, 664, 383, 575, 384, 752, 386, 580, 387, 680, 388, 581, 389, 734, 395, 592, 391, 688, 392, 590, 396, 771, 402, 604, 397, 700, 403, 606, 405, 757, 408, 608, 407, 709, 409, 614, 410, 810, 411, 617, 414, 723, 415, 624, 416, 780, 417, 628, 419, 735, 420, 632, 423, 821, 427, 640, 428, 747, 429, 643, 430, 804, 431, 649, 433, 761, 434, 653, 437, 863, 438, 658, 439, 772, 441, 662, 440, 830, 443, 667, 447, 783, 448, 673, 449, 870, 451, 677, 452, 792, 454, 681, 455, 854, 457, 689, 456, 803, 464, 694, 458, 911, 459, 692, 460, 811, 465, 699, 467, 879, 469, 706, 473, 829, 475, 711, 477, 923, 478, 717, 479, 839, 482, 722, 480, 902, 483, 726, 484, 849, 486, 728, 487, 958, 490, 736, 492, 862, 494, 741, 495, 931, 496, 746, 498, 875, 500, 748, 499, 969, 501, 754, 505, 886, 508, 762, 509, 954, 510, 766, 512, 897, 515, 773, 516, 1023,...

 

 

 

[100 terms of T]

 

 

 

[1000 terms of T]

 

 

 

[10000 terms of T]

 

 

 

 

[20000 terms of T]

 

Many thanks to both of you, Jean-Marc and Hans!

(and sorry for the double take – but you know, getting old is dealing with horrible memory problems! – Worse, I’ve recently discovered that I’m having problems with my short term memory and my short term memory!-)