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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.

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
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

It is understood that a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.
In other words, this is the lexicographically earliest nondecreasing sequence of positive numbers which is equal to its RUNS transform. - N. J. A. Sloane, Nov 07 2018
Also called Silverman's sequence.
Vardi gives several identities satisfied by A001463 and this sequence.
We can interpret this sequence as a triangle: start with 1; 2,2; 3,3; and proceed by letting the row sum of row m-1 be the number of elements of row m. The partial sums of the row sums give 1, 5, 11, 38, 272, ... Conjecture: this proceeds as Lionel Levine's sequence A014644. See also A113676. - Floor van Lamoen, Nov 06 2005
A Golomb-type sequence, that is, one with the property of being a sequence of run length of itself, can be built over any sequence with distinct terms by repeating each term a corresponding number of times, in the same manner as a(n) is built over natural numbers. See cross-references for more examples. - Ivan Neretin, Mar 29 2015
From Amiram Eldar, Jun 19 2021: (Start)
Named after the American mathematician Solomon Wolf Golomb (1932-2016).
Guy (2004) called it "Golomb's self-histogramming sequence", while in previous editions of his book (1981 and 1994) he called it "Silverman's sequence" after David Silverman. (End)
a(n) is also the number of numbers that occur n times. - Leo Crabbe, Feb 15 2025

Examples

			a(1) = 1, so 1 only appears once. The next term is therefore 2, which means 2 appears twice and so a(3) is also 2 but a(4) must be 3. And so on.
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 4*x^8 + ... - _Michael Somos_, Nov 07 2018

References

  • Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 10.
  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E25, p. 347-348.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001463 (partial sums) and A262986 (start of first run of length n).
First differences are A088517.
Golomb-type sequences over various substrates (from Glen Whitney, Oct 12 2015):
A000002 and references therein (over periodic sequences),
A109167 (over nonnegative integers),
A080605 (over odd numbers),
A080606 (over even numbers),
A080607 (over multiples of 3),
A169682 (over primes),
A013189 (over squares),
A013322 (over triangular numbers),
A250983 (over integral sums of itself).
Applying "ee Rabot" to this sequence gives A319434.
See also A095114.

Programs

  • Haskell
    a001462 n = a001462_list !! (n-1)
a001462_list = 1 : 2 : 2 : g 3  where
   g x = (replicate (a001462 x) x) ++ g (x + 1)
-- Reinhard Zumkeller, Feb 09 2012
  • Magma
    [ n eq 1 select 1 else 1+Self(n-Self(Self(n-1))) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Maple
    N:= 10000: A001462[1]:= 1: B[1]:= 1: A001462[2]:= 2:
for n from 2 while B[n-1] <= N do
  B[n]:= B[n-1] + A001462[n];
  for j from B[n-1]+1 to B[n] do A001462[j]:= n end do
end do:
seq(A001462[j],j=1..N); # Robert Israel, Oct 30 2012
  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Table[ a[n], {n, 84}] (* Robert G. Wilson v, Aug 26 2005 *)
GolSeq[n_]:=Nest[(k = 0; Flatten[# /. m_Integer :> (ConstantArray[++k,m])]) &, {1, 2}, n]
GolList=Nest[(k = 0;Flatten[# /.m_Integer :> (ConstantArray[++k,m])]) &, {1, 2},7]; AGolList=Accumulate[GolList]; Golomb[n_]:=Which[ n <= Length[GolList], GolList[[n]], n <= Total[GolList],First[FirstPosition[AGolList, ?(# > n &)]], True, $Failed] (* _JungHwan Min, Nov 29 2015 *)
  • PARI
    a = [1, 2, 2]; for(n=3, 20, for(i=1, a[n], a = concat(a, n))); a /* Michael Somos, Jul 16 1999 */
  • PARI
    {a(n) = my(A, t, i); if( n<3, max(0, n), A = vector(n); t = A[i=2] = 2; for(k=3, n, A[k] = A[k-1] + if( t--==0, t = A[i++]; 1)); A[n])}; /* Michael Somos, Oct 21 2006 */
  • Python
    a=[0, 1, 2, 2]
for n in range(3, 21):a+=[n for i in range(1, a[n] + 1)]
a[1:] # Indranil Ghosh, Jul 05 2017

Formula

a(n) = phi^(2-phi)*n^(phi-1) + E(n), where phi is the golden number (1+sqrt(5))/2 (Marcus and Fine) and E(n) is an error term which Vardi shows is O( n^(phi-1) / log n ).
a(1) = 1; a(n+1) = 1 + a(n+1-a(a(n))). - Colin Mallows
a(1)=1, a(2)=2 and for a(1) + a(2) + ... + a(n-1) < k <= a(1) + a(2) + ... + a(n) we have a(k)=n. - Benoit Cloitre, Oct 07 2003
G.f.: Sum_{n>0} a(n) x^n = Sum_{k>0} x^a(k). - Michael Somos, Oct 21 2006
a(A095114(n)) = n and a(m) < n for m < A095114(n). - Reinhard Zumkeller, Feb 09 2012 [First inequality corrected from a(m) < m by Glen Whitney, Oct 06 2015]
Conjecture: a(n) >= n^(phi-1) for all n. - Jianing Song, Aug 19 2021
a(n) = A095114(n+1) - A095114(n). - Alan Michael Gómez Calderón, Dec 21 2024 after Ralf Stephan

A113722 A variant of Golomb's sequence using odd numbers: a(n) is the number of times 2*n+1 occurs, starting with a(1) = 1.

Original entry on oeis.org

1, 3, 5, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27
Offset: 0

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Comments

a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.

Examples

			Start with 1 in row 1 and form a triangle where row n is generated from row n-1 by the rule given in the description. Then row 2 will have (1) 3, row 3 will have (3) 5's, row 4 will have (5) 7's, (5) 9's and (5) 11's, etc.
The triangle begins:
1;
3;
5,5,5;
7,7,7,7,7,9,9,9,9,9,11,11,11,11,11; ...
The number of terms in each row (also row sums with offset) is given by A113723: [1,1,3,15,135,3845,769605,3821696361,...].

Crossrefs

Cf. A001462 (Golomb's sequence), A113723, A113724, A113676.

Programs

  • PARI
    a=[1,3,5,5,5];for(n=3,20, for(i=1,a[n],a=concat(a,2*n+1)));a

A113723 The number of terms in row n of A113722 when interpreted as a triangle.

Original entry on oeis.org

1, 1, 3, 15, 135, 3845, 769605, 3821696361
Offset: 1

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113722 is a variant of Golomb's sequence using odd numbers: a(n) is the number of times 2*n+1 occurs, starting with a(1) = 1.

Crossrefs

Cf. A001462 (Golomb's sequence), A113722, A113724, A113725, A113676.

A113724 A variant of Golomb's sequence using even numbers: a(n) is the number of times 2*n+2 occurs, starting with a(1) = 2.

Original entry on oeis.org

2, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26
Offset: 1

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Comments

a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.

Examples

			Start with 2 in row 1 and form a triangle where row n is generated from row n-1 by the rule given in the description. Then row 2 will have (2) 4's, row 3 will have (4) 6's and (4) 8's, etc.
The triangle begins:
2;
4,4;
6,6,6,6,8,8,8,8; ...
The number of terms in each row (also row sums with offset) is given by A113725: [1,2,8,56,984,87848,115679160,...].

Crossrefs

Cf. A001462 (Golomb's sequence), A113725, A113722, A113676.
Cf. A080606. [From R. J. Mathar, Aug 13 2008]

Programs

  • PARI
    a=[2,4,4];for(n=2,20, for(i=1,a[n],a=concat(a,2*n+2)));a

A113725 The number of terms in row n of A113724 when interpreted as a triangle.

Original entry on oeis.org

1, 2, 8, 56, 984, 87848, 115679160
Offset: 1

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113724 is a variant of Golomb's sequence using even numbers: a(n) is the number of times 2*n+2 occurs, starting with a(1) = 2.

Crossrefs

Cf. A001462 (Golomb's sequence), A113722, A113723, A113724, A113676.

A113730 A variant of Golomb's sequence interpreted as a triangle: Start with a(1)=1 as first row, filled from left to right; fill row n+1 with the natural numbers continuing from row n in alternating order, with multiplicity according to the numbers given in row n in alternating order.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 5, 4, 4, 4, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 38, 38, 38, 38, 38, 38, 37, 37, 37, 37, 37, 37, 36, 36, 36, 36, 36, 36, 35, 35, 35, 35, 35, 35, 34, 34, 34, 34, 34, 34, 34, 33, 33, 33, 33, 33, 33, 33, 32, 32, 32, 32
Offset: 1

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Examples

			The triangle begins
1;
2;
3,3;
5,5,5,4,4,4;
6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11;
38,38,38,38,38,38,...,13,12,12,12,12,12,12,12,12,12,12,12;
From row 4 [5,5,5,4,4,4] which was filled from right to left we fill row 4
from left to right with (5) times a 6 and (5) times a 7, (5) times an 8, (4)
times a 9, (4) times a 10 and (4) times an 11.

Crossrefs

Cf. A001462 (Golomb's sequence), A113676, A113731, A113734.

A113731 Number of terms in row n of A113730 when interpreted as a triangle.

Original entry on oeis.org

1, 1, 2, 6, 27, 225, 5273, 684632, 1490340903
Offset: 1

Views

Author

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

Keywords

Comments

a(n+1) is also the row sum of row n of A113730.

Crossrefs

Cf. A001462 (Golomb's sequence), A113723, A113725, A113676.
First differences of A113734.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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