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%I A001462 M0257 N0091
%S A001462 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,
%T A001462 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,
%U A001462 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
%N A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with 
               a(1) = 1.
%C A001462 It is understood that a(n) is taken to be the smallest number >= a(n-1) 
               which is compatible with the description.
%C A001462 Also called Silverman's sequence.
%C A001462 Vardi gives several identities satisfied by A001463 and this sequence.
%C A001462 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, fvlamoen(AT)hotmail.com, Nov 06 2005.
%C A001462 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 S. Plouffe in his 1992 dissertationd is wrong. 
               - N. J. A. Sloane (njas(AT)research.att.com), May 13 2008
%D A001462 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; p. 10.
%D A001462 S. W. Golomb, Problem 5407, Amer. Math. Monthly, 73 (1966), 674.
%D A001462 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 66.
%D A001462 J. Grytczuk, Another variation on Conway's recursive sequence, Discr. 
               Math. 282 (2004), 149-161.
%D A001462 R. K. Guy, Unsolved Problems in Number Theory, E25.
%D A001462 D. Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly 
               74 (1967), 740-743.
%D A001462 Petermann, Y.-F. S., On Golomb's self-describing sequence, J. Number 
               Theory 53 (1995), 13-24.
%D A001462 Petermann, Y.-F. S., On Golomb's self-describing sequence, II, Arch. 
               Math. (Basel) 67 (1996), 473-477.
%D A001462 Petermann, Y.-F. S., Is the error term wild enough? Analysis (Munich) 
               18 (1998), 245-256.
%D A001462 Petermann, Y.-F. S. and Remy, Jean-Luc, Golomb's self-described sequence 
               and functional-differential equations, Illinois J. Math. 42 (1998), 
               420-440.
%D A001462 J. L. Remy, J. Number Theory, 66 (1997), 1-28.
%D A001462 J. Sauerberg and L. Shu, The long and the short on counting sequences, 
               Amer. Math. Monthly, 104 (1997), 306-317.
%D A001462 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001462 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.
%D A001462 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001462 I. Vardi, The error term in Golomb's sequence, J. Number Theory, 40 (1992), 
               1-11. (See also the Math. Review, 93d:11103)
%H A001462 T. D. Noe, <a href="b001462.txt">Table of n, a(n) for n = 1..10000</a>
%H A001462 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Numerical analogues of Aronson's sequence</
               a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A001462 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://arXiv.org/
               abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a> 
               (math.NT/0305308)
%H A001462 Y.-F. S. Petermann, J.-L. Remy and I. Vardi, <a href="http://www.lix.polytechnique.fr/
               ~ilan/discrete_derivatives.ps">Discrete derivatives of sequences</
               a>, Adv. in Appl. Math. 27 (2001), 562-84.
%H A001462 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001462 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001462 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A001462 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">
               Seven Staggering Sequences</a>.
%H A001462 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SilvermansSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A001462 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001462 <a href="Sindx_Aa.html#aan">Index entries for sequences of the a(a(n)) 
               = 2n family</a>
%F A001462 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 ).
%F A001462 a(1) = 1; a(n+1) = 1 + a(n+1-a(a(n))). - C. L. Mallows.
%F A001462 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 (benoit7848c(AT)orange.fr), Oct 07 
               2003
%F A001462 G.f.: Sum_{k>0} x^a(k). - Michael Somos Oct 21 2006
%e A001462 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.
%p A001462 t1 := [ 1,2,2 ]: for i from 3 to 20 do t2 := t1; for j from 1 to t1[ 
               i ] do t2 := [ op(t2),i ]; od: t1 := t2; od: t1; A001462 := n->t1[n];
%t A001462 a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Table[ a[n], {n, 84}] 
               (from Robert G. Wilson v (rgwv(at)rgwv.com), Aug 26 2005)
%o A001462 (PARI) a=[ 1,2,2 ]; for(n=3,20, for(i=1,a[ n ],a=concat(a,n))); a
%o A001462 (PARI) {a(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 */
%o A001462 (MAGMA) [ n eq 1 select 1 else 1+Self(n-Self(Self(n-1))) : n in [1..100] 
               ]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%Y A001462 Cf. A000002, A001463 (partial sums).
%Y A001462 Sequence in context: A126848 A067085 A055086 this_sequence A082462 A005041 
               A030530
%Y A001462 Adjacent sequences: A001459 A001460 A001461 this_sequence A001463 A001464 
               A001465
%K A001462 easy,nonn,nice,core
%O A001462 1,2
%A A001462 N. J. A. Sloane (njas(AT)research.att.com).

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