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A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.
(Formerly M0257 N0091)
58
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Also called Silverman's sequence.

Vardi gives several identities satisfied by A001463 and this sequence.

We can interpret A001462 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 Levile's sequence A014644. See also A113676. - Floor van Lamoen, Nov 06 2005.

The g.f. -z*(-1+z**4+z**7-z**8+z**9-z**3-z-z**11+z**12)/(1+z)/(z**2+1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong. - N. J. A. Sloane, May 13 2008

a(A095114(n)) = n and a(m) < m for m < A095114(n). - Reinhard Zumkeller, Feb 09 2012

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

REFERENCES

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 10.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.

R. K. Guy, Unsolved Problems in Number Theory, E25.

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

T. D. Noe, Table of n, a(n) for n = 1..10000

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

S. W. Golomb, Problem 5407, Amer. Math. Monthly, 73 (1966), 674.

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Brady Haran and Tony Padilla, Six Sequences - Numberphile (2013).

D. Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly 74 (1967), 740-743.

Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.

Y.-F. S. Petermann, On Golomb's self-describing sequence, J. Number Theory 53 (1995), 13-24.

Y.-F. S. Petermann, On Golomb's self-describing sequence, II, Arch. Math. (Basel) 67 (1996), 473-477.

Y.-F. S. Petermann, Is the error term wild enough?, Analysis (Munich) 18 (1998), 245-256.

Y.-F. S. Petermann, and Jean-Luc Remy, Golomb's self-described sequence and functional-differential equations, Illinois J. Math. 42 (1998), 420-440.

Y.-F. S. Petermann, J.-L. Remy and I. Vardi, Discrete derivatives of sequences, Adv. in Appl. Math. 27 (2001), 562-84.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

J. L. Remy, Sur la suite autodécrite de Golomb, J. Number Theory, 66 (1997), 1-28.

J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Seven Staggering Sequences.

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)

Eric Weisstein's World of Mathematics, Silverman's Sequence

Index entries for "core" sequences

Index entries for sequences of the a(a(n)) = 2n family

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_{k>0} x^a(k). - Michael Somos, Oct 21 2006

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.

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 *)

PROG

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

(PARI) A001462(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])} /* Michael Somos, Oct 21 2006 */

(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

(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

CROSSREFS

Cf. A001463 (partial sums).

Golomb-type sequences over various substrates:

A000002 and references therein (over periodic sequences),

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).

Sequence in context: A232753 A067085 A055086 * A082462 A005041 A030530

Adjacent sequences:  A001459 A001460 A001461 * A001463 A001464 A001465

KEYWORD

easy,nonn,nice,core

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 2 14:56 EDT 2015. Contains 261284 sequences.