A001462 -id:A001462 - cp's OEIS frontend

A113676 Number of elements of rows of Golomb's sequence A001462, with one less 2, interpreted as triangle: Start with first row 1. The row sum of row n-1 gives the number of elements taken from A001642 (one less 2) of row n.

1, 1, 2, 6, 27, 234, 6202, 1084009, 4362192095

Offset: 1

Floor van Lamoen and Paul D. Hanna, Nov 06 2005

a(n+1) gives row sum of row n of this triangle.
Conjecture: a(n) for n>1 gives first differences of Lionel Levine's sequence A014644(n) for n>=3.
Conjecture: Final elements of the rows form A014644 except for duplicate 2.

			The triangle begins
  1;
  2;
  3,3;
  4,4,4,5,5,5;
  ...
Row 4: [4,4,4,5,5,5] is generated from row 3: [3,3] because there are (3) 4's and (3) 5's in row 4.

A001463 Partial sums of A001462; also a(n) is the last occurrence of n in A001462.

Original entry on oeis.org

1, 3, 5, 8, 11, 15, 19, 23, 28, 33, 38, 44, 50, 56, 62, 69, 76, 83, 90, 98, 106, 114, 122, 131, 140, 149, 158, 167, 177, 187, 197, 207, 217, 228, 239, 250, 261, 272, 284, 296, 308, 320, 332, 344, 357, 370, 383, 396, 409, 422, 436, 450, 464, 478, 492, 506, 521, 536, 551, 566, 581, 596

Offset: 1

N. J. A. Sloane

Vardi gives several identities satisfied by A001463 and this sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly 74 (1967), 740-743.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. L. Rémy, Sur la suite autodécrite de Golomb, J. Number Theory, vol. 66 1997 pp. 1-28.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
I. Vardi, The error term in Golomb's sequence, J. Number Theory, 40 (1992), 1-11. (See also the Math. Review, 93d:11103)

a001463 n = a001463_list !! (n-1)
a001463_list = scanl1 (+) a001462_list
-- Reinhard Zumkeller, Apr 28 2012

Accumulate[a[1]=1;a[n_]:=a[n]=1+a[n-a[a[n-1]]];Table[a[n],{n,84}]] (* Harvey P. Dale, Oct 20 2011, from Robert G. Wilson v's program in A001463 *)

a(n) is asymptotic to tau^(1-tau)*n^tau where tau is the golden ratio, tau=(1+sqrt(5))/2. More precisely, a(n)= tau^(1-tau)*n^tau + c*n^(1/tau)+0(n^(1/tau)) where c is a constant around 0.6. Is there any known value for c? - Benoit Cloitre, Oct 29 2002

A109512 Integers which are not the sum of n and A001462(n).

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 17, 22, 27, 32, 38, 44, 50, 57, 64, 71, 78, 86, 94, 102, 110, 119, 128, 137, 146, 156, 166, 176, 186, 196, 207, 218, 229, 240, 251, 263, 275, 287, 299, 311, 324, 337, 350, 363, 376, 389, 403, 417, 431, 445, 459, 473, 488, 503, 518, 533, 548

Offset: 0

Eric Angelini, Alexandre Wajnberg and Robert G. Wilson v, Aug 29 2005

Terms computed by Robert G. Wilson v.

			Let N=succession of the natural numbers, G=Golomb self-describing sequence, S=sum of n and according G(n):
N=1.2.3.4.5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20.21.22
G=1,2,2,3,3,.4,.4,.4,.5,.5,.5,.6,.6,.6,.6,.7,.7,.7,.7,.8,.8,.8
S=2.4.5.7.8.10.11.12.14.15.16.18.19.20.21.23.24.25.26.28.29.30
Integers not in S (0,1,3,6,9,13,17,22...) form the sequence.

Robert Price, Table of n, a(n) for n = 0..9999

a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Do[ a[n], {n, 10^5}]; h = Complement[ Range[100000 + a[10^5]], Table[n + a[n], {n, 10000}]] t = Table[n + Sum[PrimePi[k], {k, 1, n}], {n, 0, 1200}]

A169864 The sequence S of a pair S, T generalizing Golomb's sequence A001462 and the pair A093848, A169863. See Comments for definition.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 12, 13, 15, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 91, 93

Offset: 1

N. J. A. Sloane, Jun 27 2010, following a posting from Eric Angelini to the Sequence Fans Mailing List, Jun 22, 2010

S is built with the rules [1] that you cannot insert any integer between two "touching" runs (run 1,3 is followed by 4,6,8,10,12; there is no integer between 3 and 4) and [2] that S and T share no integer.
T encodes the instructions for S.
It is not quite clear to me if T is the complement of S, or is merely disjoint from it.

Cf. A169865, A001462, A093848, A169863, A169866, A169867.

A169865 The sequence T of a pair S, T generalizing Golomb's sequence A001462 and the pair A093848, A169863. See Comments for definition.

Original entry on oeis.org

2, 5, 7, 9, 11, 14, 16, 18, 20, 22, 24, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46

Offset: 1

N. J. A. Sloane, Jun 27 2010, following a posting from Eric Angelini to the Sequence Fans Mailing List, Jun 22, 2010

S is built with the rules [1] that you cannot insert any integer between two "touching" runs (run 1,3 is followed by 4,6,8,10,12; there is no integer between 3 and 4) and [2] that S and T share no integer.

Cf. A169864, A001462, A093848, A169863.

A319434 Take Golomb's sequence A001462 and shorten all the runs by 1.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19

Offset: 1

N. J. A. Sloane, Oct 02 2018

nonn

In other words, apply Lenormand's "raboter" transformation (see A318921) to A001462.
Each value of n (n >= 2) appears exactly A001462(n)-1 times.
There should be a simple formula for a(n), just as there is for Golomb's sequence. - N. J. A. Sloane, Nov 15 2018. After 10000 terms, a(n) seems to be growing like constant*n^0.640. - N. J. A. Sloane, Jun 04 2021

			Golomb's sequence begins 1, 2,2, 3,3, 4,4,4, 5,5,5, ...
and we just shorten each run by one term, getting 2, 3, 4,4, 5,5, ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Brady Haran and N. J. A. Sloane, Planing Sequences (Le Rabot), Numberphile video, June 2021.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A001462, A318921, A319951.

a(n) = A001462(A001462(A001462(n) + n) + n). - Alan Michael Gómez Calderón, Aug 14 2025

More terms from Rémy Sigrist, Oct 04 2018

A321020 A hybrid of Kolakoski's sequence A000002 and Golomb's sequence A001462: if A001462(n) is odd replace it with 1, if even with 2.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

N. J. A. Sloane, Nov 11 2018

nonn

This is A000002 rewritten so the run lengths are given by A001462.
The companion sequence, A001462 rewritten so the run lengths are given by A000002, seems to be A156253.
Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating.

Rémy Sigrist, Table of n, a(n) for n = 1..25000
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A000002, A001462, A156253.

a = vector(84, k, k); for (i=1, oo, for (j=1, a[i], a[n++] = i; print1 (2-(i%2) ", "); if (n==#a, break(2)))) \\ Rémy Sigrist, Nov 12 2018

A377862 A variant of Golomb's sequence (A001462): the n-th digit of the sequence gives the number of times n appears, with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19

Offset: 1

Rémy Sigrist, Nov 10 2024

This sequence is a base-10 variant of A167500.
Numbers corresponding to positions of zeros do not appear in the sequence.
This sequence first differ from A001462 for n = 169: a(169) = 31 whereas A001462(169) = 29.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 500 of this sequence and of A001462
Rémy Sigrist, PARI program

Cf. A001462, A087739, A167500, A377863 (missing numbers), A377896.

PARI
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A306230 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, G(n) <> G(a(n)) (where G denotes the Golomb's sequence A001462).

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 9, 10, 7, 8, 12, 11, 16, 17, 18, 13, 14, 15, 20, 19, 24, 25, 26, 21, 22, 23, 29, 30, 27, 28, 34, 35, 36, 31, 32, 33, 39, 40, 37, 38, 45, 46, 47, 48, 41, 42, 43, 44, 51, 52, 49, 50, 57, 58, 59, 60, 53, 54, 55, 56, 63, 64, 61, 62, 70, 71, 72

Offset: 1

Rémy Sigrist, Jan 30 2019

nonn

This sequence is a self-inverse permutation of the natural numbers.

For any n > 0, let b(n) = a(n) - n; the sequence b is unbounded.

			The first terms of the sequence, alongside G, are:
  n   a(n)  G(n)
  --  ----  ----
   1     2     1
   2     1     2
   3     4     2
   4     3     3
   5     6     3
   6     5     4
   7     9     4
   8    10     4
   9     7     5
  10     8     5
  11    12     5
  12    11     6
  13    16     6
  14    17     6
  15    18     6
  16    13     7
  17    14     7
  18    15     7
  19    20     7
  20    19     8

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A306230
Index entries for sequences that are permutations of the natural numbers

See A306229 for a similar sequence.

Cf. A001462.

PARI
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See Links section.
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A143124 Triangle read by rows, sum {j=k..n}, A001462(j), 1<=k<=n, A001462 = Golomb's sequence.

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 7, 5, 3, 11, 10, 8, 6, 3, 15, 14, 12, 10, 7, 4, 19, 18, 16, 14, 11, 8, 4, 23, 22, 20, 18, 15, 12, 8, 4, 28, 27, 25, 23, 20, 17, 13, 9, 5, 33, 32, 30, 28, 25, 22, 1814, 10, 5, 38, 37, 35, 3330, 27, 23, 19, 15, 10, 5, 44, 43, 41, 39, 36, 33, 29, 25, 21, 16, 11, 6

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 26 2008

Right border of the triangle = Golomb's sequence, A014262.
Left border = A001463.
Row sums = A143125: (1, 5, 11, 23, 38, 62, 90, 122,...).

			First few rows of the triangle =
1;
3, 2;
5, 4, 2;
8, 7, 5, 3;
11, 10, 8, 6, 3;
15, 14, 12, 10, 7, 4;
19, 18, 16, 14, 11, 8, 4;
23, 22, 20, 18, 15, 12, 8, 4;
28, 27, 25, 23, 20, 17, 13, 9, 5;
...
T(5,3) = 8 = (3 + 3 + 2) where Golomb's sequence = (1, 2, 2, 3, 3, 4, 4, 4,...).

Cf. A001462, A001463, A143125.

Triangle read by rows, T(n,k) = sum {j=k..n} A001462(j), 1<=k<=n; where A001462 = (1, 2, 2, 3, 3, 4, 4,...). A000012 * (A001462 * 0^(n-k)) * A000012

cp's OEIS Frontend

A113676 Number of elements of rows of Golomb's sequence A001462, with one less 2, interpreted as triangle: Start with first row 1. The row sum of row n-1 gives the number of elements taken from A001642 (one less 2) of row n.

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A001463 Partial sums of A001462; also a(n) is the last occurrence of n in A001462.

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A109512 Integers which are not the sum of n and A001462(n).

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A169864 The sequence S of a pair S, T generalizing Golomb's sequence A001462 and the pair A093848, A169863. See Comments for definition.

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A169865 The sequence T of a pair S, T generalizing Golomb's sequence A001462 and the pair A093848, A169863. See Comments for definition.

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A319434 Take Golomb's sequence A001462 and shorten all the runs by 1.

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A321020 A hybrid of Kolakoski's sequence A000002 and Golomb's sequence A001462: if A001462(n) is odd replace it with 1, if even with 2.

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A377862 A variant of Golomb's sequence (A001462): the n-th digit of the sequence gives the number of times n appears, with a(1) = 1 and a(2) = 2.

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A306230 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, G(n) <> G(a(n)) (where G denotes the Golomb's sequence A001462).

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A143124 Triangle read by rows, sum {j=k..n}, A001462(j), 1<=k<=n, A001462 = Golomb's sequence.

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