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

A095114 a(1)=1. a(n) = a(n-1) + (number of elements of {a(1),...,a(n-1)} that are <= n-1).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 24, 29, 34, 39, 45, 51, 57, 63, 70, 77, 84, 91, 99, 107, 115, 123, 132, 141, 150, 159, 168, 178, 188, 198, 208, 218, 229, 240, 251, 262, 273, 285, 297, 309, 321, 333, 345, 358, 371, 384, 397, 410, 423, 437, 451, 465, 479, 493, 507, 522
Offset: 1

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Author

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

Keywords

Comments

Every positive integer is either of the form a(n)+n-1 or of the form a(n+1)-a(n)+n, but not both.
The sequence a(n)+n-1 is A109512. - Robert Price, Apr 16 2013
The sequence a(n+1)-a(n)+n is A224731. - Robert Price, Apr 16 2013
Equals A001463 + 1, the partial sums of Golomb's sequence A001462. - Ralf Stephan, May 28 2004
a(n) is the position of the first occurrence of n in A001462, i.e., A001462(a(n)) = n and A001462(m) < n for m < a(n). - Reinhard Zumkeller, Feb 09 2012 [Explanation added and first inequality corrected from A001462(m) < m by Glen Whitney, Oct 06 2015]

Examples

			3 elements of {a(1),...,a(4)} are <= 4, so a(5) = a(4) + 3 = 9.

Links

Crossrefs

Equals A001463(n) + 1.
Cf. A109512, A224731.

Programs

  • Haskell
    a095114 n = a095114_list !! (n-1)
a095114_list = 1 : f [1] 1 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + length [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012
  • Maple
    a[1]:= 1; m:= 0;
for n from 2 to 100 do
  if a[m+1] <= n-1 then m:= m+1 fi;
  a[n]:= a[n-1]+m;
od:
seq(a[i],i=1..100); # Robert Israel, Oct 07 2015
  • Mathematica
    a[1]=1; a[n_]:=a[n]=a[n-1]+Length[Select[a/@Range[n-1], #
  • PARI
    a(n) = sum(k=1, n-1, t(k)) + 1;
t(n)=local(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]);
vector(100, n, a(n)) \\ Altug Alkan, Oct 06 2015

A088517 First differences of Golomb's sequence.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Benoit Cloitre, Nov 14 2003

Keywords

Comments

Characteristic function of A001463

Links

Programs

  • Haskell
    a088517 n = a001462 (n + 1) - a001462 n
a088517_list = zipWith (-) (tail a001462_list) a001462_list
-- Reinhard Zumkeller, Apr 28 2012

Formula

a(n)=A001462(n+1)-A001462(n); a(A001463(n))=1

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

Views

Author

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

Keywords

Comments

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

Examples

			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,...).

Crossrefs

Cf. A001462, A001463, A143125.

Formula

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

A109336 "Que sera, sera" sequence: self-describing sequence where a(n) gives the number of n+1's which will be concatenated to form a(n+1); starting with a(1) = 1.

Original entry on oeis.org

1, 2, 33, 444444444444444444444444444444444
Offset: 1

Views

Author

Alexandre Wajnberg, Aug 23 2005

Keywords

Examples

			a(1) says: there will be one 2 in a(2).
a(2)=2 because a(1) said so; and a(2)=2 says: there will be two 3's in a(3).
a(3)=33 because a(2) said so; and also a(3) says: there will be thirty three 4's in a(4).
Therefore a(4)= 444444444444444444444444444444444 (33 times the digit 4).
And a(5)= 555555555555555...555 (with 444444444444444444444444444444444 5's).

Crossrefs

Cf. A001462, A001463, A103320, A102357, A076782.

Formula

a(1) = 1. For n > 1, let k = floor(1+log_10(n)); then a(n) = n*(10^(k*a(n-1))-1)/(10^k-1).

Extensions

Formula corrected to handle n>9 also by Rick L. Shepherd, Mar 22 2009
Showing 1-5 of 5 results.

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

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