Search: id:A003714
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%I A003714
%S A003714 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,
%T A003714 72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,
%U A003714 148,149,160,161,162,164,165,168,169,170,256,257,258,260
%N A003714 Fibbinary numbers: if n = F_i1+F_i2+...+F_ik is the Zeckendorf representation 
               of n (i.e. write n in Fibonacci number system) then a(n) = 2^{i1-2}+2^{i2-2}+...+2^{ik-2}.
%C A003714 The name "Fibbinary" is due to Marc LeBrun (mlb(AT)well.com)
%C A003714 "... integers whose binary representation contains no consecutive ones 
               and noticed that the number of such numbers with n bits was fibonacci(n)" 
               posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net) 
               Jul 17 2002.
%C A003714 n is in the sequence if and only if C(3n,2n) is odd; also a(n) (mod 2) 
               = A003849(n) - Benoit Cloitre, Mar 8 2003
%C A003714 Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 03 2005. This implies that A003188(2*a(n)) = 3*a(n) holds for 
               all n.
%C A003714 A116361(a(n)) <= 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 04 2006
%C A003714 Numbers n>=1 such that base 2 representation contains no block of 1's 
               longer than 1. For example, n=17 in binary 10001 belongs to the sequence, 
               but n=19 in binary 10011 is not in the sequence as it contains 11 
               which is a block of 1's longer then 1. - Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               May 13 2008
%C A003714 n is in the sequence if and only if the central Stirling number of the 
               second kind S(2n,n)=A007820(n) is odd. [From O-Yeat Chan (math(AT)oyeat.com), 
               Sep 03 2009]
%D A003714 O-Yeat Chan, http://www.oyeat.com/papers/stirling9.pdf [From O-Yeat Chan 
               (math(AT)oyeat.com), Sep 03 2009]
%D A003714 D. E. Knuth, Art of Comp. Programming, Vol. 1, 2nd ed., pp. 85, 493.
%H A003714 T. D. Noe, <a href="b003714.txt">Table of n, a(n) for n=0..1363</a>
%H A003714 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A003714 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/
               ~allouche/kimb.ps">Self-generating sets, integers with missing blocks 
               and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A003714 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrep.html">Rabbit Sequence in Zeckendorf Expansion (A003714)</a>
%H A003714 <a href="Sindx_Con.html#CongruCrossDomain">Index entries for sequences 
               defined by congruent products between domains N and GF(2)[X]</a>
%H A003714 <a href="Sindx_Con.html#CongruXOR">Index entries for sequences defined 
               by congruent products under XOR</a>
%F A003714 No two adjacent 1's in binary expansion.
%F A003714 Let f(x) := sum(x^Fibbinary(n), n, 0, inf). Then f satisfies the functional 
               equation f(x) = x f(x^4) + f(x^2).
%F A003714 a(0)=0, a(1)=1, a(2)=2, a(n) = 2^(A072649(n)-1) + a(n - A000045(1+A072649(n))) 
               - Antti Karttunen
%F A003714 It appears that sequence gives n such that A082759(3n) is odd; or, probably 
               equivalently, n such that A037011(3n)=1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 20 2003
%F A003714 If n is in the sequence then so are 2n and 4n+1. - Henry Bottomley (se16(AT)btinternet.com), 
               Jan 11 2005
%p A003714 with(combinat, fibonacci); A003714 := proc(n) option remember; if(n < 
               3) then RETURN(n); else RETURN((2^(A072649(n)-1))+A003714(n-fibonacci(1+A072649(n)))); 
               fi; end;
%t A003714 f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t 
               = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; 
               t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[ fr, 2]]; 
               Table[ f[n], {n, 0, 61}] (from Robert G. Wilson v Sep 18 2004)
%Y A003714 A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. 
               Char. function: A085357. Even terms: A022340, Odd terms: A022341.
%Y A003714 Other sequences based on similar restrictions on binary expansion: A003754, 
               A048715, A048718, A107907, A107909.
%Y A003714 Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, 
               A060112, A072649, A083368, A089939, A106027, A106028, A116361.
%Y A003714 3*a(n) is in A001969.
%Y A003714 Sequence in context: A045702 A005658 A166021 this_sequence A010402 A010443 
               A035269
%Y A003714 Adjacent sequences: A003711 A003712 A003713 this_sequence A003715 A003716 
               A003717
%K A003714 nonn,nice
%O A003714 0,3
%A A003714 N. J. A. Sloane (njas(AT)research.att.com).
%E A003714 Edited Feb 21 2006 by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com).
%E A003714 Cross reference to A007820 added (into O-Y.C. comment) by Jason Kimberley 
               (Jason.Kimberley(AT)newcastle.edu.au), Sep 14 2009

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