A030124 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For any n, all integers k satisfying sum(i=1,n,a(i))+1<k<sum(i=1,n+1,a(i))+1 are in the sequence. E.g., sum(i=1,3,a(i))+1=12, sum(i=1,4,a(i))+1=18, hence 13,14,15,16,17 are in the sequence. - Benoit 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"

FORMULA

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

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 *)

PROG

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

(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

CROSSREFS