Commas variants
The
story of the magnificent video made by my son Lorenzo for my birthday can be
seen here (in French). If you want to see this video now, click on the “Lorenzo Angelini” link ("Happy Birthday" – added yesterday by Neil S. in the LINKS section).
Watching
the 17-year-old Commas sequence again, I suddenly had the idea of a few
variants – Double Comma, Triple Comma, Quadruple Comma and
Reverse Comma.
If you
want to bring these new sequences to life and co-sign them with me in the
OEIS, I would be delighted! Please check the numbers below, compute a few more and submit the seqs to
the OEIS.
NAME
The
"double commas" sequence: the lexicographically earliest sequence of
positive numbers with the property that the sequence formed by the pairs of
digits adjacent to the commas between the terms is twice the sequence of
successive differences between the terms.
DATA
1, 25,
127, 271, 295, 403, 471, 499, 691, 725, 841, 877, 1019, 1201, 1223, 1285, 1387,
1529, 1711, 1733, 1795, 1897, 2041, 2065, 2169, 2353, 2417, 2561, 2585, 2689,
2873, 2937, 3083, 3149, ...
EXAMPLE
a(2) – a(1)
= 24 and 24 is twice the 12 visible around the 1st comma: [1,25]
a(3) – a(2)
= 102 and 102 is twice the 51 visible around the 2nd comma: [25,127]
a(4) – a(3)
= 144 and 144 is twice the 72 visible around the 3rd comma: [127,271]
a(5) – a(4)
= 24 and 24 is twice the 12 visible around the 4th comma: [271,295]; etc.
__________
Next day update
Giorgos Kalogeropoulos has checked and extended the sequences – many thanks Giorgos, good job!
> x2
> All terms confirmed. Here are the first 70 terms
1, 25, 127, 271, 295, 403, 471, 499, 691, 725, 841, 877, 1019, 1201, 1223, 1285, 1387, 1529, 1711, 1733, 1795, 1897, 2041, 2065, 2169, 2353, 2417, 2561, 2585, 2689, 2873, 2937, 3083, 3149, 3335, 3441, 3467, 3613, 3679, 3865, 3971, 3997, 4145, 4253, 4321, 4349, 4537, 4685, 4793, 4861, 4889, 5079, 5269, 5459, 5649, 5839, 6031, 6063, 6135, 6247, 6399, 6591, 6623, 6695, 6807, 6959, 7153, 7227, 7381, 7415
NAME
The
"triple commas" sequence: the lexicographically earliest sequence of
positive numbers with the property that the sequence formed by the pairs of
digits adjacent to the commas between the terms is thrice the sequence of
successive differences between the terms.
DATA
1, 43, 136,
325, 487, 718, 985, 1138, 1381, 1414, 1537, 1750, 1753, 1846, 2032, 2098, 2344,
2470, 2476, 2662, 2728, 2974, 3103, 3202, 3271, 3310, 3319, 3598, 3847, 4069,
4351, 4393, 4495, 4657, …
EXAMPLE
a(2) – a(1)
= 42 and 42 is thrice the 14 visible around the 1st comma: [1,43]
a(3) – a(2)
= 93 and 93 is thrice the 31 visible around the 2nd comma: [43,136]
a(4) – a(3)
= 189 and 189 is thrice the 63 visible around the 3rd comma: [136,325]
a(5) – a(4)
= 162 and 162 is thrice the 54 visible around the 4th comma: [325,487]; etc.
__________
GK:
> x3
> All terms confirmed. Here are the first 70 terms
1, 43, 136, 325, 487, 718, 985, 1138, 1381, 1414, 1537, 1750, 1753, 1846, 2032, 2098, 2344, 2470, 2476, 2662, 2728, 2974, 3103, 3202, 3271, 3310, 3319, 3598, 3847, 4069, 4351, 4393, 4495, 4657, 4879, 5164, 5299, 5584, 5719, 6007, 6235, 6403, 6511, 6559, 6847, 7078, 7339, 7630, 7651, 7702, 7783, 7894, 8038, 8302, 8386, 8590, 8614, 8758, 9025, 9202, 9289, 9586, 9793, 9910, 9937, 10150, 10153, 10246, 10429, 10702
NAME
The
"quadruple commas" sequence: the lexicographically earliest sequence
of positive numbers with the property that the sequence formed by the pairs of
digits adjacent to the commas between the terms is four times the sequence of
successive differences between the terms.
DATA
1, 65,
273, 409, 797, 1081, 1125, 1329, 1693, 1817, 2105, 2313, 2441, 2489, 2857,
3149, 3428, 3760, 3772, 3864, 4040,4056, 4312, 4408, 4744, 4920, 4932, 5032,
5132, 5232, 5332, 5432, 5532, 5632, 5732, 5832, 5932, 6036, …
EXAMPLE
a(2) – a(1)
= 64 and 64 is four times the 16 visible around the 1st comma: [1,65]
a(3) – a(2)
= 208 and 208 is four times the 52 visible around the 2nd comma: [65,273]
a(4) – a(3)
= 136 and 136 is four times the 34 visible around the 3rd comma: [273,409]
a(5) – a(4)
= 388 and 388 is four times the 97 visible around the 4th comma: [409,797]; etc.
__________
GK:
> x4
> All terms confirmed. Here are the first 70 terms
1, 65, 273, 409, 797, 1081, 1125, 1329, 1693, 1817, 2105, 2313, 2441, 2489, 2857, 3149, 3521, 3573, 3705, 3917, 4213, 4349, 4725, 4941, 4997, 5297, 5597, 5897, 6201, 6265, 6489, 6873, 7021, 7089, 7477, 7785, 8017, 8329, 8721, 8793, 8945, 9181, 9257, 9573, 9729, 10093, 10217, 10501, 10545, 10749, 11113, 11237, 11521, 11565, 11769, 12133, 12257, 12541, 12585, 12789, 13153, 13277, 13561, 13605, 13809, 14173, 14297, 14581, 14625, 14829
NAME
The
"backwards commas" sequence: the lexicographically earliest sequence
of positive numbers with the property that the sequence formed by the reversed pairs
of digits adjacent to the commas between the terms is the same as the sequence
of successive differences between the terms.
DATA
1, 92,
104, 118, 136, 152, 164, 178, 206, 232, 254, 278, 316, 352, 384, 428, 476, 532,
584, 648, 726, 812, 894, 988, 1006, 1022, 1034, 1048, 1066, 1082, 1094, 1108,
1126, 1142, 1154, 1168, 1186, 1202, 1214, …
EXAMPLE
a(2) – a(1)
= 91 and 91 is 19 read backwards, which is visible around the 1st
comma: [1,92]
a(3) – a(2)
= 12 and 12 is 21 read backwards, which is visible around the 2nd comma:
[92,104]
a(4) – a(3)
= 14 and 14 is 41 read backwards, which is visible around the 3rd comma:
[104,118]
a(5) – a(4)
= 18 and 18 is 81 read backwards, which is visible around the 4th comma:
[118,136];
etc.
__________
Giorgos Kalogeropoulos:
> For the sequence with the reversed digits I have some questions:
> Why can't the second term be 12 or even 22?
> I know that the sequence terminates with these numbers but shouldn't there be a rule to exclude them?
> Also, if the second term is indeed 92 then my program returns a different sequence:
1, 92, 104, 118, 136, 152, 164, 178, 196, 222, 244, 268, 296, 332, 364, 398, 446, 492, 544, 598, 666, 742, 824, 918
> It confirms only the first 8 terms and then terminates at 25th term.
Eric Angelini:
(...) Yes, Giorgos, you are right about the last « Reversed Commas » seq which starts genuinely only with 1, 92 and not with 12 or 22 — I should have mentioned that but I was exhausted (I am not even 100% sure of the beginning 1, 92 as being the lexicographically earliest not halting immediately). If your finite seq is the good one, I think it would be of interest to submit it — because it might give « reversed ideas » to other people:
> 1, 92, 104, 118, 136, 152, 164, 178, 196, 222, 244, 268, 296, 332, 364, 398, 446, 492, 544, 598, 666, 742, 824, 918.
Question: can’t we backtrack this last seq? And try to extend it with more terms than above? Like this, for instance:
> 1, 92, 104, 118, 136, 152, 164, 178, 206, 232, 254, etc.
Best,
É.
____________________
Equinox update
GK:
> Hi Eric, I tried some things and here are my results...
The thing with this sequence is that in each step you have one or more paths to choose from.
Your sequence with 92 and 206 (which seem arbitrary) terminates at the 250th term.
Then I thought about making the complete graph, and when I did, I discovered that this graph with all these different paths is finite.
The longest path of this graph has 2514 terms
And finally I discovered a simple rule that leads to this path:
In each step you choose the biggest possible number!
To get an idea here are the first terms with the "next possible terms"
1->{12,22,32,42,52,62,72,82,92}
92->{104}
104->{118}
118->{136}
136->{152}
152->{164}
164->{178}
178->{196,206}
206->{232}
232->{254}
254->{278}
278->{316}
316->{352}
352->{384}
384->{428}
428->{476}
476->{532}
532->{584}
584->{648}
648->{726}
726->{812}
812->{894,904}
904->{998}
998->{1016}
1016->{1032}
1032->{1044}
1044->{1058}
1058->{1076}
1076->{1092}
1092->{1104}
....
So, we get a sequence from 1 to 99952 with 2514 terms which I'm sending you in a txt file.
In each step, finding the candidates is pretty easy as you have to choose the possible numbers with the 9 starting different digits.
Here are also all the points that make new paths in this particular path.
1 -> {12,22,32,42,52,62,72,82,92}
178 -> {196,206}
812 -> {894,904}
1976 -> {1992,2002}
3956 -> {3992,4002}
19984 -> {19998,20008}
39958 -> {39996,40006}
49952 -> {49994,50004}
EA:
Wonderful data, many thanks Giorgos!
__________________________
(txt file)
{1,92,104,118,136,152,164,178,206,232,254,278,316,352,384,428,476,532,584,648,726,812,904,998,1016,1032,1044,1058,1076,1092,1104,1118,1136,1152,1164,1178,1196,1212,1224,1238,1256,1272,1284,1298,1316,1332,1344,1358,1376,1392,1404,1418,1436,1452,1464,1478,1496,1512,1524,1538,1556,1572,1584,1598,1616,1632,1644,1658,1676,1692,1704,1718,1736,1752,1764,1778,1796,1812,1824,1838,1856,1872,1884,1898,1916,1932,1944,1958,1976,2002,2024,2048,2076,2102,2124,2148,2176,2202,2224,2248,2276,2302,2324,2348,2376,2402,2424,2448,2476,2502,2524,2548,2576,2602,2624,2648,2676,2702,2724,2748,2776,2802,2824,2848,2876,2902,2924,2948,2976,3012,3044,3078,3116,3152,3184,3218,3256,3292,3324,3358,3396,3432,3464,3498,3536,3572,3604,3638,3676,3712,3744,3778,3816,3852,3884,3918,3956,4002,4044,4088,4136,4182,4224,4268,4316,4362,4404,4448,4496,4542,4584,4628,4676,4722,4764,4808,4856,4902,4944,4988,5046,5102,5154,5208,5266,5322,5374,5428,5486,5542,5594,5648,5706,5762,5814,5868,5926,5982,6044,6108,6176,6242,6304,6368,6436,6502,6564,6628,6696,6762,6824,6888,6956,7032,7104,7178,7256,7332,7404,7478,7556,7632,7704,7778,7856,7932,8014,8098,8186,8272,8354,8438,8526,8612,8694,8778,8866,8952,9044,9138,9236,9332,9424,9518,9616,9712,9804,9898,9996,10012,10024,10038,10056,10072,10084,10098,10116,10132,10144,10158,10176,10192,10204,10218,10236,10252,10264,10278,10296,10312,10324,10338,10356,10372,10384,10398,10416,10432,10444,10458,10476,10492,10504,10518,10536,10552,10564,10578,10596,10612,10624,10638,10656,10672,10684,10698,10716,10732,10744,10758,10776,10792,10804,10818,10836,10852,10864,10878,10896,10912,10924,10938,10956,10972,10984,10998,11016,11032,11044,11058,11076,11092,11104,11118,11136,11152,11164,11178,11196,11212,11224,11238,11256,11272,11284,11298,11316,11332,11344,11358,11376,11392,11404,11418,11436,11452,11464,11478,11496,11512,11524,11538,11556,11572,11584,11598,11616,11632,11644,11658,11676,11692,11704,11718,11736,11752,11764,11778,11796,11812,11824,11838,11856,11872,11884,11898,11916,11932,11944,11958,11976,11992,12004,12018,12036,12052,12064,12078,12096,12112,12124,12138,12156,12172,12184,12198,12216,12232,12244,12258,12276,12292,12304,12318,12336,12352,12364,12378,12396,12412,12424,12438,12456,12472,12484,12498,12516,12532,12544,12558,12576,12592,12604,12618,12636,12652,12664,12678,12696,12712,12724,12738,12756,12772,12784,12798,12816,12832,12844,12858,12876,12892,12904,12918,12936,12952,12964,12978,12996,13012,13024,13038,13056,13072,13084,13098,13116,13132,13144,13158,13176,13192,13204,13218,13236,13252,13264,13278,13296,13312,13324,13338,13356,13372,13384,13398,13416,13432,13444,13458,13476,13492,13504,13518,13536,13552,13564,13578,13596,13612,13624,13638,13656,13672,13684,13698,13716,13732,13744,13758,13776,13792,13804,13818,13836,13852,13864,13878,13896,13912,13924,13938,13956,13972,13984,13998,14016,14032,14044,14058,14076,14092,14104,14118,14136,14152,14164,14178,14196,14212,14224,14238,14256,14272,14284,14298,14316,14332,14344,14358,14376,14392,14404,14418,14436,14452,14464,14478,14496,14512,14524,14538,14556,14572,14584,14598,14616,14632,14644,14658,14676,14692,14704,14718,14736,14752,14764,14778,14796,14812,14824,14838,14856,14872,14884,14898,14916,14932,14944,14958,14976,14992,15004,15018,15036,15052,15064,15078,15096,15112,15124,15138,15156,15172,15184,15198,15216,15232,15244,15258,15276,15292,15304,15318,15336,15352,15364,15378,15396,15412,15424,15438,15456,15472,15484,15498,15516,15532,15544,15558,15576,15592,15604,15618,15636,15652,15664,15678,15696,15712,15724,15738,15756,15772,15784,15798,15816,15832,15844,15858,15876,15892,15904,15918,15936,15952,15964,15978,15996,16012,16024,16038,16056,16072,16084,16098,16116,16132,16144,16158,16176,16192,16204,16218,16236,16252,16264,16278,16296,16312,16324,16338,16356,16372,16384,16398,16416,16432,16444,16458,16476,16492,16504,16518,16536,16552,16564,16578,16596,16612,16624,16638,16656,16672,16684,16698,16716,16732,16744,16758,16776,16792,16804,16818,16836,16852,16864,16878,16896,16912,16924,16938,16956,16972,16984,16998,17016,17032,17044,17058,17076,17092,17104,17118,17136,17152,17164,17178,17196,17212,17224,17238,17256,17272,17284,17298,17316,17332,17344,17358,17376,17392,17404,17418,17436,17452,17464,17478,17496,17512,17524,17538,17556,17572,17584,17598,17616,17632,17644,17658,17676,17692,17704,17718,17736,17752,17764,17778,17796,17812,17824,17838,17856,17872,17884,17898,17916,17932,17944,17958,17976,17992,18004,18018,18036,18052,18064,18078,18096,18112,18124,18138,18156,18172,18184,18198,18216,18232,18244,18258,18276,18292,18304,18318,18336,18352,18364,18378,18396,18412,18424,18438,18456,18472,18484,18498,18516,18532,18544,18558,18576,18592,18604,18618,18636,18652,18664,18678,18696,18712,18724,18738,18756,18772,18784,18798,18816,18832,18844,18858,18876,18892,18904,18918,18936,18952,18964,18978,18996,19012,19024,19038,19056,19072,19084,19098,19116,19132,19144,19158,19176,19192,19204,19218,19236,19252,19264,19278,19296,19312,19324,19338,19356,19372,19384,19398,19416,19432,19444,1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