

A001462


Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.
(Formerly M0257 N0091)


147



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
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OFFSET

1,2


COMMENTS

It is understood that a(n) is taken to be the smallest number >= a(n1) 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 m1 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 Levile's sequence A014644. See also A113676.  Floor van Lamoen, Nov 06 2005
A Golombtype 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 crossreferences for more examples.  Ivan Neretin, Mar 29 2015
Named after the American mathematician Solomon Wolf Golomb (19322016).
Guy (2004) called it "Golomb's selfhistogramming sequence", while in previous editions of his book (1981 and 1994) he called it "Silverman's sequence" after David Silverman. (End)


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. AddisonWesley, Reading, MA, 1990, p. 66.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E25, p. 347348.
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. 93110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Solomon W. Golomb, Problem 5407, Amer. Math. Monthly, Vol. 73, No. 6 (1966), p. 674.
Brady Haran and Tony Padilla, Six Sequences, Numberphile video (2013).
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)


FORMULA

a(n) = phi^(2phi)*n^(phi1) + 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^(phi1) / log n ).
a(1)=1, a(2)=2 and for a(1) + a(2) + ... + a(n1) < 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
Conjecture: a(n) >= n^(phi1) for all n.  Jianing Song, Aug 19 2021


EXAMPLE

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


MAPLE

for n from 2 while B[n1] <= N do
for j from B[n1]+1 to B[n] do A001462[j]:= n end do
end do:


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 *)


PROG

(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[k1] + if( t==0, t = A[i++]; 1)); A[n])}; /* Michael Somos, Oct 21 2006 */
(Magma) [ n eq 1 select 1 else 1+Self(nSelf(Self(n1))) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergeihaller.de), Dec 21 2006
(Haskell)
a001462 n = a001462_list !! (n1)
a001462_list = 1 : 2 : 2 : g 3 where
g x = (replicate (a001462 x) x) ++ g (x + 1)
(Python)
a=[0, 1, 2, 2]
for n in range(3, 21):a+=[n for i in range(1, a[n] + 1)]


CROSSREFS

Cf. A001463 (partial sums) and A262986 (start of first run of length n).
Golombtype sequences over various substrates (from Glen Whitney, Oct 12 2015):
A000002 and references therein (over periodic sequences),
A109167 (over nonnegative integers),
A250983 (over integral sums of itself).
Applying "ee Rabot" to this sequence gives A319434.


KEYWORD

easy,nonn,nice,core


AUTHOR



STATUS

approved



