Hello
Math-Fun & SeqFan,
WARNING!
This
sequence is *not* for OEIS mathematicians!
- it is base-dependent!
- it has black spots on its shirt!
- it is badly defined (in pidgin-English)!
- it is computed by hand...
(I
would recommend it only for drunken ophthalmologists.)
...
Now proceed to 1-click reading © at
your own risks!
-------------------------------------------------------
1,12,35,94,135,186,248,331,344,387,461,475,530,535,590,595,651,667,744,791,809,908,997,1068,1149,1240,1241,1252,...
The
difference between a(n) and a(n-1) is always a 2-digit
figure -- which is reproduced to the left and right of the separating comma:
Sequence: 1, 12, 35, 94, 135, 186, 248, 331,
344, 387, 461, 475, 530, 535, ...
Differences: 11 23
59 41 51
62 83 13
43 74 14
55 05 ...
You
will agree that “05” is not the common way to represent a difference of five
units (in yellow)... Well, so it is here!
Could
someone check if the sequence grows infinitely?
[If you
start the sequence with 3 instead of 1 you’ll get blocked very quickly: 3,36 -- END.]
[Self-blocking
integers are: 18,27,36,45,54,63,72 and 81]
I
could trace only four chains leading to such dead-ends:
3,36 END
31,45 END
15,72 END
43,81 END.
---------------------------------------------------------
I’ve
examined too a few sequences showing the |absolute difference| between a(n) and a(n-1).
I
had thus more choices for a(n+1). So I decided to
build a sequence with differences < 100 and differences > 9 (in
order to avoid leading zeros). I wanted also the sequence to be kept as low as
possible -- and not self-blocking -- and not self-looping either...
Well,
I’m not sure all those criteria are compatible between them and define properly
sequences like this one:
1,12,35,94,135,78,159,251,239,148,62,91,74,115,166,105,156,88,4,48,129,221,209,118,199...
Anyway,
this was great fun to investigate!
Best,
É.
---------------------------------------------------------
The
above message was posted by me on rec.puzzles as well. I got this answer from “The Qurqirish
Dragon” almost immediately:
I noticed one problem with the rule:
Take the sequence starting with 2:
2, (+22) 24, (+47) 71, ?
Is the next value 89 or 90? Both follow the given
rule:
2, (+22) 24, (+47) 71, (+18) 89, (+91) 180, ...
2, (+22) 24, (+47) 71, (+19) 90, (+09) 99, (+91) 190, ...
This problem will happen whenever a given addition
works resulting in a number consisting of all 9s (except for the first digit,
which must be less than 9).
I
got then the same remark from Nicolas Graner, a friend of mine I had written too:
Que fais-tu si tu rencontres 14, 33, 52 ou 71 qui ont deux
successeurs
possibles ?
[What do you do with 14, 33, 52 or 71 which have two
possible successors?]
To
both I wrote:
Well
done! I overlooked that possibility -- please keep the sequence as low as
possible!
Then
I got this from Zak Seidov
-- another friend:
For checking purposes I give first and 10th hundreds
terms of seq:
1,12,35,94,135,186,248,331,344,387,461,475,530,535,590,595,651,667,744,791,809,908,997,
1068,1149,1240,1241,1252,1273,1304,1345,1396,1457,1528,1609,1700,1701,1712,1733,1764,
1805,1856,1917,1988,2070,2072,2094,2136,2198,2280,2282,2304,2346,2408,2490,2492,2514,
2556,2618,2700,2702,2724,2766,2828,2910,2912,2934,2976,3039,3132,3155,3208,3291,3304,
3347,3420,3423,3456,3519,3612,3635,3688,3771,3784,3827,3900,3903,3936,3999,4093,4127,
4201,4215,4269,4363,4397,4471,4485,4539,4633,
<...>
44303,44337,44411,44425,44479,44573,44607,44681,44695,44749,44843,44877,44951,44965,
45019,45113,45147,45221,45235,45289,45383,45417,45491,45505,45559,45653,45687,45761,
45775,45829,45923,45957,46031,46045,46099,46193,46227,46301,46315,46369,46463,46497,
46571,46585,46639,46733,46767,46841,46855,46909,47003,47037,47111,47125,47179,47273,
47307,47381,47395,47449,47543,47577,47651,47665,47719,47813,47847,47921,47935,47989,
48083,48117,48191,48205,48259,48353,48387,48461,48475,48529,48623,48657,48731,48745,
48799,48893,48927,49001,49015,49069,49163,49197,49271,49285,49339,49433,49467,49541,
49555,49609,
Zak
Then
I got this, from Edwin Clark:
Eric,
If I didn't make a mistake in my Maple program:
The last term in your sequence --call it a(n) is
a(2137453)=99999945;
---but there is no next term.
Best wishes,
Edwin
Waow! How quick, Edwin!
Zak answered him (on SeqFan):
Edwin,
This dirty code
a=49703;c=1001;Do[ida=a+10Mod[a,10];Do[b=ida+i;If[i==IntegerDigits[b][[1]],a=b;c++;Break[],If[i9,Print[a];Goto[en]]],{i,0,9}],{10000000}];Label[en];Print[{c,a,b,"end!"}]
99999945
{2137453,99999945,100000004,end!}
confirms your great result!
Ed Murphy sent to rec.puzzles a
message about “blocked sequences”:
a(1) is chosen arbitrarily.
a(n+1) can be any number satisfying the following conditions:
1) d = a(n+1)-a(n) < 100
2) a(n) mod 10 = floor(d/10)
3) d mod 10 = floor(a(n+1)/10^floor(log10(a(n+1))))
In other words, the difference between a(n) and a(n+1)
is a two-digit number that starts with the same digit as the last digit of a(n)
(or is < 10 if a(n) ends in 0), and ends with the same digit as the first
digit of a(n+1).
A given number may have 0, 1, or >1 candidate
successors.
Sample chain:
1, 12, 35,
94, 135, 186, 248, 331, 344, 387, 461, 475, 530, 535, 590
11 23 59
41 51 62
83 13 43
74 14 55
5 55
Sample chains that become blocked:
3, 36
31, 45
15, 72
43, 81
Which choices for a(1) result
in a block? Here's a first stab:
10 for s =
1 to 100
20 dim c[100]; c[1] = s, cc = 1
30 for i = 1 to 100000
40 dim nc[100]; np = 0
50 for ci = 1 to cc
60 for d
= mod(c[ci],10)*10+1 to
mod(c[ci],10)*10+9
70 d$ =
str(d),
n$ = str(c[ci]+d)
80 if mid(d$,-1,1) = mid(n$,1,1) then np
+= 1, nc[np] = c[ci]+d
90 next d
100 next ci
110 if np = 0 then print s, " blocks
on iteration ", i; break
120 c{all} = nc{all}, cc = np
130 next i
140 next s
3 blocks on iteration 2
7 blocks on iteration 15
15 blocks on iteration 2
18 blocks on iteration 1
19 blocks on iteration 21482
21 blocks on iteration 2074
25 blocks on iteration 193
27 blocks on iteration 1
28 blocks on iteration 197
31 blocks on iteration 2
34 blocks on iteration 2073
36 blocks on iteration 1
37 blocks on iteration 196
39 blocks on iteration 20
41 blocks on iteration 19
42 blocks on iteration 1981
43 blocks on iteration 2
45 blocks on iteration 1
49 blocks on iteration 193
53 blocks on iteration 21
54 blocks on iteration 1
56 blocks on iteration 18
60 blocks on iteration 204
63 blocks on iteration 1
65 blocks on iteration 2100
66 blocks on iteration 203
68 blocks on iteration 1980
72 blocks on iteration 1
74 blocks on iteration 20
75 blocks on iteration 2136
81 blocks on iteration 1
82 blocks on iteration 2072
83 blocks on iteration 192
85 blocks on iteration 14
86 blocks on iteration 1983
90 blocks on iteration 21
92 blocks on iteration 20
98 blocks on iteration 205
99 blocks on iteration 20
Alas, Qurqirish Dragon
is wrong; reaching a three-digit sum does not guarantee that the sequence can
be extended indefinitely.
Example:
7, 85, 136, 197, 269, 362, 385, 439, 534, 579, 675,
732, 759, 857, 936
Notice how the number of iterations before a block
clump around 2, 20, 200, 2000, 20000.
Look at that sequence starting with 7:
From 136 to 936, the differences increase roughly linearly within the
11-99 range, averaging somewhere around 50; the amount of increase is
approximately 1000, so the number of increases is approximately 1000/50 =
20. The sequences that block after
approximately 200 iterations are probably hitting similar tar pits at 9xxx
-> 1xxxx, and so forth.
ObPuzzle: What is the largest number of
choices we can have at any iteration?
IOW, how large do the C and NC arrays need to be to guarantee they won't
overflow? Adding the following lines:
125 if m < np then m = np
150 print m
indicates that the maximum number this program actually encounters within 100
iterations is 4, within 1,000 is 5, and within 100,000 is 6.
What if d = abs(a(n+1)+a(n)),
i.e. the sequence is not necessarily monotonically increasing? Which choices for a(1)
result in a block? Which choices result in a loop?
Sample sequence:
1, 12, 35,
94, 135, 78, 159, 251, 239, 148, 62, 91, 74, 115, 166,
105, 156, 88,
4, 48, 129, 221, 209, 118, 199
What if d must be > 10? Adding the following line:
55 if
mod(c[ci],10) = 0 then
continue
indicates that only a few numbers survive to the 20-iteration hump, and none of
them get very far past it:
18 -> 5, 11, 17, 23, 25, 38, 55, 58, 61, 65, 75,
76, 78, 83, 86, 96
19 -> 5, 11, 17, 23, 25, 38, 55, 58, 61, 65,
75, 78, 96
20 -> 5, 11, 17, 23, 58, 61, 78, 96
21 -> 5, 11, 17, 23, 61, 96
22 -> 5, 11, 17
23 -> nothing
What
about bases other than 10?
Indeed,
Ed!
End
of the story -- Zak and Neil have turned this into an OEIS sequence: A121805
Thanks
to all contributors,
É.
---------------------------------------------------------
A
year later (dec. 12th, 2007) David Wilson posted this on Seq-Fans:
Every positive integer n has at most one comma sequence predecessor p(n),
that is, a number which
may precede it in a comma sequence. p(n) is
defined for all
positive integers except
for the 50 integers in the
following set
S
= {
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
13, 14, 15, 16, 17, 18, 19, 20, 21, 25,
26, 27, 28, 29, 30, 31, 32, 37, 38, 39,
40, 41, 42, 43, 49, 50, 51, 52, 53, 54,
62, 63, 64, 65, 74, 75, 76, 86, 87, 98
}
p(n) is undefined for n in S, for all other
n in N, p(n) < n.
Starting with any
positive integer n, we can iterate p on n until we
reach a unique element a(n) in S, called the ancestor of n.
Going forward, most positive integers n have a unique successor n.
However, there are a smattering of integers that have no successor,
specifically, the integers in the set
T
= { (10^x + 9y) - 100 : x >= 2 and 2 <= y <=
9 }
That is
T
= {
18, 27, 36, 45, 54, 63, 72, 81,
918, 927, 936, 945, 954, 963, 972, 981,
9918, 9927, 9936, 9945, 9954, 9963, 9972, 9981,
...
}
If a comma sequence
reaches an element of T,
the sequence ends there.
There is also a sparse
set of integers that have two successors, these
form the strangely similar set
B
= { (10^x + 9)y - 100 ; x >= 2 and 2 <= y <=
9 }
or
B
= {
118, 227, 336, 445, 554, 663, 772, 881,
1918, 2927, 3936, 4945, 5954, 6963, 7972, 8981,
19918, 29927, 39936, 49945, 59954, 69963, 79972, 89981,
...
}
(10^x + 9)y - 100
has the two successors
(10^x)y - 1 and (10^x)y. No
positive integer has more than two successors.
If a comma sequence
reaches an element of B, it is possible to continue
the sequence in two ways from
that element.
You will notice that comma sequence terminating and branching elements
all occur just
before (within 100 of) an
initial digit change. On intervals
where the initial
digit of its elements remains constant, however, a
comma
sequence behaves very nicely,
with periodic first differences. This allows
one to do some modular magic to skip over these intervals in constant time,
which allows us to quickly compute a(n) for very large n.
Using this knowledge,
I wrote a program to compute
m(n), the largest
possible element in a comma sequence starting with n, for n in S.
It turns
out that there are infinite
comma sequences starting with 20, a comma
sequence with any other
start value in S terminates
at or before 10^365-82.
n m(n)
1 10^25-28
2 10^16-82
3 10^2-64
4 10^7-55
5 10^16-64
6 10^42-64
7 10^3-64
8 10^13-28
9 10^24-19
10 10^13-19
13 10^87-37
14 10^11-82
15 10^2-28
16 10^14-55
17 10^54-55
18 10^2-82
19 10^6-19
20 Infinity
21 10^5-19
25 10^4-37
26 10^7-28
27 10^2-73
28 10^4-73
29 10^8-64
30 10^365-82
31 10^2-55
32 10^9-82
37 10^4-82
38 10^17-37
39 10^3-82
40 10^30-82
41 10^3-46
42 10^5-73
43 10^2-19
49 10^4-55
50 10^18-73
51 10^18-55
52 10^21-19
53 10^3-28
54 10^2-46
62 10^7-19
63 10^2-37
64 10^132-19
65 10^5-37
74 10^3-55
75 10^5-46
76 10^24-46
86 10^5-82
87 10^12-82
98 10^4-64
Great
results -- thanks, David, for the 20-infinite
comma sequence!
---------------------------------------
1001
terms computed by Zak Seidov.
[the difference between terms n and (n+1) is “uv”, u
being the last digit of n and v the first digit of n+1).
Example:
for n=28 and (n+1)=29 we have 1252-1273=21]
# Zak Seidov, Dec
11, 2006
1 1
2 12
3 35
4 94
5 135
6 186
7 248
8 331
9 344
10 387
11 461
12 475
13 530
14 535
15 590
16 595
17 651
18 667
19 744
20 791
21 809
22 908
23 997
24
1068
25
1149
26
1240
27
1241
28
1252
29
1273
30
1304
31
1345
32
1396
33
1457
34
1528
35
1609
36
1700
37
1701
38
1712
39
1733
40
1764
41
1805
42
1856
43
1917
44
1988
45
2070
46
2072
47
2094
48
2136
49
2198
50
2280
51
2282
52
2304
53
2346
54
2408
55
2490
56
2492
57
2514
58
2556
59
2618
60
2700
61
2702
62
2724
63
2766
64
2828
65
2910
66
2912
67
2934
68 2976
69
3039
70
3132
71
3155
72
3208
73
3291
74
3304
75
3347
76
3420
77
3423
78
3456
79
3519
80
3612
81
3635
82
3688
83
3771
84
3784
85
3827
86
3900
87
3903
88
3936
89
3999
90
4093
91
4127
92
4201
93
4215
94
4269
95
4363
96
4397
97
4471
98
4485
99
4539
100 4633
101 4667
102 4741
103 4755
104 4809
105 4903
106 4937
107 5012
108 5037
109 5112
110 5137
111 5212
112 5237
113 5312
114 5337
115 5412
116 5437
117 5512
118 5537
119 5612
120 5637
121 5712
122 5737
123 5812
124 5837
125 5912
126 5937
127 6013
128 6049
129 6145
130 6201
131 6217
132 6293
133 6329
134 6425
135 6481
136 6497
137 6573
138 6609
139 6705
140 6761
141 6777
142 6853
143 6889
144 6985
145 7042
146 7069
147 7166
148 7233
149 7270
150 7277
151 7354
152 7401
153 7418
154 7505
155 7562
156 7589
157 7686
158 7753
159 7790
160 7797
161 7874
162 7921
163 7938
164 8026
165 8094
166 8142
167 8170
168 8178
169 8266
170 8334
171 8382
172 8410
173 8418
174 8506
175 8574
176 8622
177 8650
178 8658
179 8746
180 8814
181 8862
182 8890
183 8898
184 8986
185 9055
186 9114
187 9163
188 9202
189 9231
190 9250
191 9259
192 9358
193 9447
194 9526
195 9595
196 9654
197 9703
198 9742
199 9771
200 9790
201 9799
202 9898
203 8883
204 10058
205 10139
206 10230
207 10231
208 10242
209 10263
210 10294
211 10335
212 10386
213 10447
214 10518
215 10599
216 10690
217 10691
218 10702
219 10723
220 10754
221 10795
222 10846
223 10907
224 10978
225 11059
226 11150
227 11151
228 11162
229 11183
230 11214
231 11255
232 11306
233 11367
234 11438
235 11519
236 11610
237 11611
238 11622
239 11643
240 11674
241 11715
242 11766
243 11827
244 11898
245 11979
246 12070
247 12071
248 12082
249 12103
250 12134
251 12175
252 12226
253 12287
254 12358
255 12439
256 12530
257 12531
258 12542
259 12563
260 12594
261 12635
262 12686
263 12747
264 12818
265 12899
266 12990
267 12991
268 13002
269 13023
270 13054
271 13095
272 13146
273 13207
274 13278
275 13359
276 13450
277 13451
278 13462
279 13483
280 13514
281 13555
282 13606
283 13667
284 13738
285 13819
286 13910
287 13911
288 13922
289 13943
290 13974
291 14015
292 14066
293 14127
294 14198
295 14279
296 14370
297 14371
298 14382
299 14403
300 14434
301 14475
302 14526
303 14587
304 14658
305 14739
306 14830
307 14831
308 14842
309 14863
310 14894
311 14935
312 14986
313 15047
314 15118
315 15199
316 15290
317 15291
318 15302
319 15323
320 15354
321 15395
322 15446
323 15507
324 15578
325 15659
326 15750
327 15751
328 15762
329 15783
330 15814
331 15855
332 15906
333 15967
334 16038
335 16119
336 16210
337 16211
338 16222
339 16243
340 16274
341 16315
342 16366
343 16427
344 16498
345 16579
346 16670
347 16671
348 16682
349 16703
350 16734
351 16775
352 16826
353 16887
354 16958
355 17039
356 17130
357 17131
358 17142
359 17163
360 17194
361 17235
362 17286
363 17347
364 17418
365 17499
366 17590
367 17591
368 17602
369 17623
370 17654
371 17695
372 17746
373 17807
374 17878
375 17959
376 18050
377 18051
378 18062
379 18083
380 18114
381 18155
382 18206
383 18267
384 18338
385 18419
386 18510
387 18511
388 18522
389 18543
390 18574
391 18615
392 18666
393 18727
394 18798
395 18879
396 18970
397 18971
398 18982
399 19003
400 19034
401 19075
402 19126
403 19187
404 19258
405 19339
406 19430
407 19431
408 19442
409 19463
410 19494
411 19535
412 19586
413 19647
414 19718
415 19799
416 19890
417 19891
418 19902
419 19923
420 19954
421 19995
422 20047
423 20119
424 20211
425 20223
426 20255
427 20307
428 20379
429 20471
430 20483
431 20515
432 20567
433 20639
434 20731
435 20743
436 20775
437 20827
438 20899
439 20991
440 21003
441 21035
442 21087
443 21159
444 21251
445 21263
446 21295
447 21347
448 21419
449 21511
450 21523
451 21555
452 21607
453 21679
454 21771
455 21783
456 21815
457 21867
458 21939
459 22031
460 22043
461 22075
462 22127
463 22199
464 22291
465 22303
466 22335
467 22387
468 22459
469 22551
470 22563
471 22595
472 22647
473 22719
474 22811
475 22823
476 22855
477 22907
478 22979
479 23071
480 23083
481 23115
482 23167
483 23239
484 23331
485 23343
486 23375
487 23427
488 23499
489 23591
490 23603
491 23635
492 23687
493 23759
494 23851
495 23863
496 23895
497 23947
498 24019
499 24111
500 24123
501 24155
502 24207
503 24279
504 24371
505 24383
506 24415
507 24467
508 24539
509 24631
510 24643
511 24675
512 24727
513 24799
514 24891
515 24903
516 24935
517 24987
518 25059
519 25151
520 25163
521 25195
522 25247
523 25319
524 25411
525 25423
526 25455
527 25507
528 25579
529 25671
530 25683
531 25715
532 18032
533 25839
534 25931
535 25943
536 25975
537 26027
538 26099
539 26191
540 26203
541 26235
542 26287
543 26359
544 26451
545 26463
546 26495
547 26547
548 26619
549 26711
550 26723
551 26755
552 26807
553 26879
554 26971
555 26983
556 27015
557 27067
558 27139
559 27231
560 27243
561 27275
562 27327
563 27399
564 27491
565 27503
566 27535
567 27587
568 27659
569 27751
570 27763
571 27795
572 27847
573 27919
574 28011
575 28023
576 28055
577 28107
578 28179
579 28271
580 28283
581 28315
582 28367
583 28439
584 28531
585 28543
586 28575
587 28627
588 28699
589 28791
590 28803
591 28835
592 28887
593 28959
594 29051
595 29063
596 29095
597 29147
598 29219
599 29311
600 29323
601 29355
602 29407
603 29479
604 29571
605 29583
606 29615
607 29667
608 29739
609 29831
610 29843
611 29875
612 29927
613 29999
614 30092
615 30115
616 30168
617 30251
618 30264
619 30307
620 30380
621 30383
622 30416
623 30479
624 30572
625 30595
626 30648
627 30731
628 30744
629 30787
630 30860
631 30863
632 30896
633 30959
634 31052
635 31075
636 31128
637 31211
638 31224
639 31267
640 31340
641 31343
642 31376
643 31439
644 31532
645 31555
646 31608
647 31691
648 31704
649 31747
650 31820
651 31823
652 31856
653 31919
654 32012
655 32035
656 32088
657 32171
658 32184
659 32227
660 32300
661 32303
662 32336
663 32399
664 32492
665 32515
666 32568
667 32651
668 32664
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...
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