

A030124


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


37



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
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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 nexttoleading 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. 3235, 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 figurefigure 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).
Eric Weisstein's World of Mathematics, Hofstadter FigureFigure 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[n1]  h[n2] + If[ MemberQ[ Array[h, n1], h[n1]  h[n2] + 1], 2, 1]; Differences[ Array[h, 69]] (* JeanFrançois 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

Cf. A005228, A030124, A037257, A037258, A037259, A061577, A140778, A129198, A129199, A100707, A093903, A005132, A006509, A081145, A099004, A225376, A225377, A225378, A225385, A225386, A225387, A225687.
Sequence in context: A276216 A226946 A039138 * A064318 A039100 A184482
Adjacent sequences: A030121 A030122 A030123 * A030125 A030126 A030127


KEYWORD

nonn


AUTHOR

Eric W. Weisstein


EXTENSIONS

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


STATUS

approved



