A030124 Complement (and also first differences) of Hofstadter's sequence A005228.

2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

For any n, all integers k satisfying sum(i=1,n,a(i))+1Benoit Cloitre, Apr 01 2002

The asymptotic equivalence a(n) ~ n follows from the fact that the values disallowed in the present sequence because they occur in A005228 are negligible, since A005228 grows much faster than A030124. The next-to-leading term in the formula is calculated from the functional equation F(x) + G(x) = x, suggested by D. Wilson (cf. reference), where F and G are the inverse functions of smooth, increasing approximations f and f' of A005228 and A030124. It seems that higher order corrections calculated from this equation do not agree with the real behavior of a(n). - M. F. Hasler, Jun 04 2008

A225850(a(n)) = 2*n, cf. A167151. - Reinhard Zumkeller, May 17 2013

References

• E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
• D. R. Hofstadter, "Gödel, Escher, Bach: An Eternal Golden Braid", Basic Books, 1st & 20th anniv. edition (1979 & 1999), p. 73.

Links

• T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n=1..10000
• Benoit Jubin, Asymptotic series for Hofstadter's figure-figure sequences, arXiv:1404.1791; J. Integer Sequences, 17 (2014), #14.7.2.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
• Eric Weisstein's World of Mathematics, Hofstadter Figure-Figure Sequence.
• D. W. Wilson, Asymptotics about A005228, post to the SeqFan mailing list (access restricted to subscribers), Jun 03 2008
• Index entries for sequences from "Goedel, Escher, Bach"

Crossrefs

Cf. A005228, A030124, A037257, A037258, A037259, A061577, A140778, A129198, A129199, A100707, A093903, A005132, A006509, A081145, A099004, A225376, A225377, A225378, A225385, A225386, A225387, A225687.

Programs

Haskell

import Data.List (delete)
a030124 n = a030124_list !! n
a030124_list = figureDiff 1 [2..] where
   figureDiff n (x:xs) = x : figureDiff n' (delete n' xs) where n' = n + x
-- Reinhard Zumkeller, Mar 03 2011

Mathematica

(* h stands for Hofstadter's sequence A005228 *) h[1] = 1; h[2] = 3; h[n_] := h[n] = 2*h[n-1] - h[n-2] + If[ MemberQ[ Array[h, n-1], h[n-1] - h[n-2] + 1], 2, 1]; Differences[ Array[h, 69]] (* Jean-François Alcover, Oct 06 2011 *)

PARI

{a=b=t=1;for(i=1,100, while(bittest(t,b++),); print1(b",");t+=1<M. F. Hasler, Jun 04 2008

Formula

a(n) = n + sqrt(2n) + o(n^(1/2)). - M. F. Hasler, Jun 04 2008 [proved in Jubin's paper].

Extensions

Changed offset to agree with that of A005228. - N. J. A. Sloane, May 19 2013