|
%I #59 Jun 24 2017 22:59:36
%S 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,
%T 32,33,34,36,37,38,39,40,41,42,43,44,46,47,48,49,50,51,52,53,54,55,57,
%U 58,59,60,61,62,63,64,65,66,67,68,70,71,72,73,74,75,76,77,78
%N Complement (and also first differences) of Hofstadter's sequence A005228.
%C 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
%C 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
%C A225850(a(n)) = 2*n, cf. A167151. - _Reinhard Zumkeller_, May 17 2013
%D E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%D D. R. Hofstadter, "Gödel, Escher, Bach: An Eternal Golden Braid", Basic Books, 1st & 20th anniv. edition (1979 & 1999), p. 73.
%H T. D. Noe and N. J. A. Sloane, <a href="/A030124/b030124.txt">Table of n, a(n) for n=1..10000</a>
%H Benoit Jubin, <a href="http://www.emis.de/journals/JIS/VOL17/Jubin/jubin2.html">Asymptotic series for Hofstadter's figure-figure sequences</a>, <a href="http://arxiv.org/abs/1404.1791">arXiv:1404.1791</a>; J. Integer Sequences, 17 (2014), #14.7.2.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H David Singmaster, <a href="/A005178/a005178.pdf">Letter to N. J. A. Sloane</a>, Oct 3 1982.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HofstadterFigure-FigureSequence.html">Hofstadter Figure-Figure Sequence.</a>
%H D. W. Wilson, <a href="http://list.seqfan.eu/cgi-bin/mailman/private/seqfan-old/2008-June/014299.html">Asymptotics about A005228</a>, post to the SeqFan mailing list (access restricted to subscribers), Jun 03 2008
%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>
%F a(n) = n + sqrt(2n) + o(n^(1/2)). - _M. F. Hasler_, Jun 04 2008 [proved in Jubin's paper].
%t (* 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 *)
%o (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
%o (Haskell)
%o import Data.List (delete)
%o a030124 n = a030124_list !! n
%o a030124_list = figureDiff 1 [2..] where
%o figureDiff n (x:xs) = x : figureDiff n' (delete n' xs) where n' = n + x
%o -- _Reinhard Zumkeller_, Mar 03 2011
%Y Cf. A005228, A030124, A037257, A037258, A037259, A061577, A140778, A129198, A129199, A100707, A093903, A005132, A006509, A081145, A099004, A225376, A225377, A225378, A225385, A225386, A225387, A225687.
%K nonn
%O 1,1
%A _Eric W. Weisstein_
%E Changed offset to agree with that of A005228. - _N. J. A. Sloane_, May 19 2013
|
