cp's OEIS Frontend

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 1-5 of 5 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

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.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

Original entry on oeis.org

3, 4, 6, 18, 450, 205439850, 241382525361273331926149714645357743772646450
Offset: 0

Views

Author

Gus Wiseman, May 16 2018

Keywords

Comments

The first entry 3 is optional so has offset 0.

Examples

			The list of multisets with Heinz numbers in the sequence is A014643. The number of k's in row n + 1 is equal to the k-th term of row n. The length of row n is A014644(n).
        3: {2}
        4: {1,1}
        6: {1,2}
       18: {1,2,2}
      450: {1,2,2,3,3}
205439850: {1,2,2,3,3,4,4,4,5,5,5}

Crossrefs

Cf. A001462, A014643, A014644, A055932, A056239, A112798, A130091, A181819, A181821, A182850-A182858, A296150, A275870, A304455, A304678.

Programs

  • Mathematica
    Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},6]

A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 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
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The row {2} could be safely prepended to this triangle. - Gus Wiseman, May 13 2018

Examples

			Triangle begins:
{1,1},
{1,2},
{1,2,2},
{1,2,2,3,3},
...

Links

Crossrefs

Rows converge to A001462.
Cf. A011784, A012257, A014643, A014644, A112798, A181819, A182850-A182858, A296150, A304455.

Programs

  • Maple
    T:= proc(n) option remember; `if`(n=0, 2, (l->
      seq(i$l[i], i=1..nops(l)))([T(n-1)]))
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, May 17 2018
  • Mathematica
    NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},8] (* Gus Wiseman, May 13 2018 *)

Extensions

More terms from Patrick De Geest

A113734 Maximum value in row n of A113730 when interpreted as a triangle.

Original entry on oeis.org

1, 2, 3, 5, 11, 38, 263, 5536, 690168, 1491031071
Offset: 1

Views

Author

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

Keywords

Crossrefs

Cf. A014644, A113731.
Showing 1-5 of 5 results.

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

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