A121805 -id:A121805 - cp's OEIS frontend

A367578 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - CT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b.

0, 1, 2, 14, 35, 78, 21, 103, 92, 53, 28, 60, 146, 145, 84, 26, 68, 134, 215, 173, 122, 91, 62, 46, 22, 84, 56, 11, 72, 55, 30, 83, 75, 38, 91, 180, 169, 168, 77, 164, 93, 44, 10, 51, 56, 41, 105, 94, 153, 112, 81, 109, 98, 197, 116, 45, 109, 58, 153, 234, 202, 160, 139, 138, 47, 131, 202, 190

Offset: 0

Scott R. Shannon, Nov 25 2023

This is a variation of Recamán's sequence A005132, where the step size is calculated from the Comma transform of the previous two terms, see A367360 and A121805. As the maximum step size is 99, it is likely that many numbers never appear. In the first 10 million terms the smallest numbers that do appear are 0,1,2,8,10,11,14,17,21,22. The first number to appear twice is 84. The terms show a broadly repetitive pattern that repeats every order of magnitude, although slight differences are still present; see the two attached images.

			a(2) = 2 as CT(a(0),a(1)) = CT(0,1) = 1, so a(2) = a(1) + 1 = 2.
a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
a(7) = 21 as CT(a(5),a(6)) = CT(35,78) = 57, so a(7) = a(6) - 57 = 21, as 21 is nonnegative and not already in the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000.
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 10000000 terms.

Cf. A367360, A005132, A121805.

A367598 Record high-points in A367356.

Original entry on oeis.org

17, 164, 490, 1472, 39819, 119228, 3225387, 9657464, 261256395, 782254580, 21161768043

Offset: 1

N. J. A. Sloane, Nov 24 2023

This is the base-3 analog of A367364.

Cf. A121805, A367355, A367356, A367364, A367599.

a(7)-a(11) from Michael S. Branicky, Nov 27 2023

A367599 Indices of record high-points in A367356.

Original entry on oeis.org

1, 6, 7, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442

Offset: 1

N. J. A. Sloane, Nov 24 2023

This is a base-3 analog of A367365. The present sequence includes the terms 2*3, 2*3^2, 2*3^6, whereas A367365 includes 4*10 and 4*10^3.

Terms a(4)-a(11) are of the form 2*3^(2*i), i > 0. - Michael S. Branicky, Nov 26 2023

Cf. A121805, A367355, A367356, A367364, A367598.

a(7)-a(11) from Michael S. Branicky, Nov 27 2023

A367612 Numbers that are the comma-child of exactly one positive number.

Original entry on oeis.org

11, 12, 22, 23, 24, 33, 34, 35, 36, 44, 45, 46, 47, 48, 55, 56, 57, 58, 59, 60, 61, 66, 67, 68, 69, 70, 71, 72, 73, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 15 2023

This is the complement of A367611.
See A367338 for definition of comma-child.
May also be called numbers that have a positive comma-predecessor.

Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A367611.

def ok(n): y = int(str(n)[0]); x = (n-y)%10; return n - y - 10*x > 0
print([k for k in range(1, 123) if ok(k)]) # Michael S. Branicky, Dec 15 2023

A367620 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term.

Original entry on oeis.org

20, 22, 46, 107, 178, 260, 262, 284, 327, 401, 415, 469, 564, 610, 616, 682, 709, 807, 885, 944, 993, 1024, 1065, 1116, 1177, 1248, 1329, 1420, 1421, 1432, 1453, 1484, 1525, 1576, 1637, 1708, 1789, 1880, 1881, 1892, 1913, 1944, 1985, 2037, 2109, 2201, 2213, 2245, 2297, 2369, 2461, 2473, 2505, 2557, 2629, 2721, 2733, 2765

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 20 2023

Discovered by David W. Wilson in 2007 (see 2016 Angelini link).
The first choice point occurs for the term after a(412987860) = 19999999918, which has two comma-children.
We do not know which choice to take at that point. We do know by König's Infinity Lemma that one or both forks will extend to infinity. The definition of this sequence requires that we choose the smallest fork that has an infinite continuation.
Update, Dec 22 2023: We now know that the start of this sequence is one of four candidates (all other possible starts having terminated). The shortest of the four possible starts has length
   8278670191169895553395510925614764265575448369172463113087634743486440833078554
In other words, we know that there are only four possibilities for the initial prefix of that length.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A367338, A367346.

A367622 Number of comma-children of n in base 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A121805, A367338 (definition), A367341 (0's), A367346 (2's).

f[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==
FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]; f/@Range[108] (* Ivan N. Ianakiev, Dec 24 2023 *)

def a(n):
    x = 10*(n%10)
    return len([y for y in range(1, 10) if str(n+x+y)[0] == str(y)])
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Dec 23 2023

A375567 Length of the "exponential comma sequence" with n as the initial term, or -1 if that sequence is infinite.

Original entry on oeis.org

-1, 3, 4, 1, 2, 3, 5, 4, 4, -1, 5, 1, 1, 4, 1, 1, 1, 1, 1, 6, 1, 11, 4, 9, 5, 1, 7, 2, 3, 1, 1, 1, 6, 3, 1, 6, 4, 1, 7, 1, 3, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 4, 7, 2, 4, 1, 2, 1, 3, 2, 1, 8, 3, 1, 6, 2, 1, 2, 2, 3, 3, 4, 3, 5, 1, 5, 3, 2, 1, 3, 2, 3, 4, 3

Offset: 1

Nicholas M. R. Frieler, Aug 19 2024

An "exponential comma sequence" is the lexicographically earliest sequence of positive integers (with some chosen initial term) with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive logarithms between the terms.

If the decimal expansion of n is 1...0, its exponential comma sequence is trivially constant and therefore infinite so that a(n) = -1. Conjecture: these are the only infinite exponential comma sequences.

			For n = 2, the next term of its exponential comma sequence is 67108864 because log_2(67108864) = 26 and this is the smallest number where the exponential comma property holds.

Kevin Ryde, Table of n, a(n) for n = 1..10000
Kevin Ryde, C Code

Cf. A121805, A374725.

C
```
/* See links. */
```

ExponentialCommaSequenceLength[n_] := Module[{seq = {n}, i = 1},
  While[True,
  Do[
    If[(IntegerDigits@Power[Last@seq, Mod[Last@commaSeq,10]*10 + j])[[1]] == j,
      seq = seq~Join~{Power[Last@seq, Mod[Last@commaSeq, 10]*10 + j]};
      Break[];];,
     {j, 1, 9}
    ];
   If[Length@seq != i + 1, Break[];];
   If[seq[[1]] == seq[[2]], Return[-1]];
   i++;
 ];
  Length@seq
 ]

A334639 Lexicographically earliest infinite sequence of distinct positive integers such that the result of the division of a(n+1) by a(n) starts with the decimal number [a.b] with a = the rightmost digit of a(n), b = the leftmost digit of a(n+1) and the decimal point = the comma between a(n) and a(n+1).

Original entry on oeis.org

2, 5, 26, 159, 1447, 10274, 45206, 280278, 2298281, 2757938, 22615092, 56537732, 118729239, 1080436075, 5942398413, 18421435081, 22105722098, 179056348994, 859470475172, 1804887997862, 4331731194869, 40718273231769, 378679941055453, 1173907817271905, 6573883776722668, 55878012102142678, 469375301657998496, 2910126870279590676, 17751773908705503124, 85208514761786414996

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 07 2020

Some light backtracking is needed sometimes to let the sequence go to infinity (especially when a new integer ends in zero: we then increase it by 1).

			The sequence starts with 2, 5, 26, 159, 1447, 10274, 45206,...
a(2) = 5 divided by a(1) = 2 is 2.5;
a(3) = 26 divided by a(2) = 5 starts with 5.2;
a(4) = 159 divided by a(3) = 26 starts with 6.1;
a(5) = 1447 divided by a(4) = 159 starts with 9.1;
a(6) = 10274 divided by a(5) = 1447 starts with 7.1;
a(7) = 45206 divided by a(6) = 10274 starts with 4.4; etc.

Eric Angelini, message to the Math-Fun mailing list on May 3rd 2020.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..133

Cf. A121805.

A334829 The sum a(n) + a(n+1) is visible around the comma that follows a(n+1). See the Comments and Example sections for details.

Original entry on oeis.org

1, 11, 23, 46, 91, 374, 6506, 8801, 53076, 18777, 18533, 73109, 16428, 95371, 117992, 133632, 516246, 4987805, 50405105, 539291005, 896961101, 4362521065, 2594821666, 9573427311, 21682489773, 12559170843, 42416606165, 49757770089, 21743762547, 15015326363, 67590889108, 26062154719, 36530438276, 25925929956

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 13 2020

The rule used here is that the rightmost digit of a(n+1) is the first digit of the sum a(n) + a(n+1), the other digits of the said sum being put after the comma in order to start a(n+2).
As no digit 0 (zero) can start a term, one will have to backtrack sometimes in order to extend the sequence - and pick another term for a(n+1), compatible with the above rule. This is always possible.
Note that the sequence is not monotonically increasing as shown by a(10) and a(11) for instance; still, the 1000th term is 406-digit long.
The sequence is always extended with the smallest available integer not yet present that does not lead to a contradiction.

			a(1) + a(2) is 1 + 11 = 12 and 12 can be seen here: 1(1,2)3,
a(2) + a(3) is 11 + 23 = 34 and 34 can be seen here: 2(3,4)6,
a(3) + a(4) is 23 + 46 = 69 and 69 can be seen here: 4(6,9)1,
a(4) + a(5) is 46 + 91 = 137 and 137 can be seen here: 9(1,37)4,
a(5) + a(6) is 91 + 374 = 465 and 465 can be seen here: 37(4,65)06, etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..1002
E. Angelini More and more commas on the SeqFan mailing list, May 12 2020.

Cf. A121805, A230450, A197756.

A367357 First differences of A134647.

Original entry on oeis.org

1, 1, 12, 21, 43, 57, 81, 52, 62, 83, 4, 34, 74, 15, 55, 5, 56, 6, 66, 27, 87, 58, 29, 9, 99, 81, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 72, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 3, 23, 53

Offset: 1

N. J. A. Sloane, Nov 20 2023

Cf. A121805, A134647.

cp's OEIS Frontend

A367578 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - CT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b.

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A367598 Record high-points in A367356.

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A367599 Indices of record high-points in A367356.

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A367612 Numbers that are the comma-child of exactly one positive number.

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A367620 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term.

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A367622 Number of comma-children of n in base 10.

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A375567 Length of the "exponential comma sequence" with n as the initial term, or -1 if that sequence is infinite.

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C

Mathematica

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A334829 The sum a(n) + a(n+1) is visible around the comma that follows a(n+1). See the Comments and Example sections for details.

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A367357 First differences of A134647.

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