Search: id:A005252
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%I A005252 M1048
%S A005252 1,1,1,1,2,4,7,11,17,27,44,72,117,189,305,493,798,1292,2091,3383,5473,
%T A005252 8855,14328,23184,37513,60697,98209,158905,257114,416020,673135,
%U A005252 1089155,1762289,2851443,4613732,7465176,12078909,19544085,31622993
%N A005252 Sum_{k=0..floor(n/4)} binomial(n-2k,2k).
%C A005252 The Twopins/t sequence (see Guy).
%C A005252 Number of closed walks of length n at a vertex of the graph with adjacency 
               matrix [1,1,0,0;0,0,0,1;1,0,0,0;0,0,1,1] - Paul Barry (pbarry(AT)wit.ie), 
               Mar 15 2004
%C A005252 a(n+3) = number of n-bit sequences that avoid both 010 and 0110. Example: 
               for n=3, there are 8 3-bit sequences and only 010 fails to qualify, 
               so a(6)=7. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004
%D A005252 R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. 
               Quart., 16 (1978), 84-86.
%D A005252 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, 
               p. 205 of first edition.
%D A005252 R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical 
               Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
%D A005252 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal 
               triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A005252 MacKay, Information Theory, Inference and Learning Algorithms, CUP, 2003, 
               p251.
%D A005252 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005252 T. D. Noe, <a href="b005252.txt">Table of n, a(n) for n=0..500</a>
%H A005252 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=424">
               Encyclopedia of Combinatorial Structures 424</a>
%H A005252 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005252 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005252 Second differences give sequence shifted twice - E. L. Tan, Univ. Phillipines.
%F A005252 G.f.: (1-x)/((1-x-x^2)(1-x+x^2)); a(n)=Fib(n+1)/2+A010892(n)/2; a(n)=(((1+sqrt(5))/
               2)^(n+1)/sqrt(5)-((1-sqrt(5))/2)^(n+1)/sqrt(5)+cos(pi*n/3)+sin(pi*n/
               3)/sqrt(3))/2. - Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
%F A005252 a(n) = 2*a(n-1)-a(n-2)+a(n-4); a(0) = a(1) = a(2) = a(3) = 1 . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), May 01 2006
%p A005252 A005252:=(-1+z)/(z**2-z+1)/(z**2+z-1); [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%p A005252 ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,X))), X = Sequence(b,
               card >= 2)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=2..40); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2008
%Y A005252 First differences of A024490.
%Y A005252 Sequence in context: A117276 A035295 A006999 this_sequence A023430 A023429 
               A023428
%Y A005252 Adjacent sequences: A005249 A005250 A005251 this_sequence A005253 A005254 
               A005255
%K A005252 nonn,easy,nice
%O A005252 0,5
%A A005252 N. J. A. Sloane (njas(AT)research.att.com).
%E A005252 More terms from (and formula corrected by) James A. Sellers (sellersj(AT)math.psu.edu), 
               Feb 06 2000
%E A005252 Definition revised by N. J. A. Sloane, Aug 16 2009 at the suggestion 
               of Alessandro Orlandi.

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