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A005252
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Sum_{k=0..floor(n/4)} binomial(n-2k,2k).
(Formerly M1048)
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12
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1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 72, 117, 189, 305, 493, 798, 1292, 2091, 3383, 5473, 8855, 14328, 23184, 37513, 60697, 98209, 158905, 257114, 416020, 673135, 1089155, 1762289, 2851443, 4613732, 7465176, 12078909, 19544085, 31622993
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OFFSET
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0,5
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COMMENTS
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The Twopins/t sequence (see Guy).
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, Mar 15 2004
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, Mar 25 2004
a(n) = A173021(2^(n-1) - 1) for n>0. [From Reinhard Zumkeller, Feb 07 2010]
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REFERENCES
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R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205 of first edition.
R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,2).
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
MacKay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p251.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 424
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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Second differences give sequence shifted twice - E. L. Tan, Univ. Phillipines.
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, Mar 15 2004
a(n) = 2*a(n-1)-a(n-2)+a(n-4); a(0) = a(1) = a(2) = a(3) = 1 . - Philippe Deléham, May 01 2006
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MAPLE
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A005252:=(-1+z)/(z**2-z+1)/(z**2+z-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
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
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MATHEMATICA
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Table[Sum[Binomial[n-2k, 2k], {k, 0, Floor[n/4]}], {n, 0, 50}] (* or *) LinearRecurrence[{2, -1, 0, 1}, {1, 1, 1, 1}, 50] (* From Harvey P. Dale, Dec 09 2011 *)
Table[HypergeometricPFQ[{1/4-n/4, 1/2-n/4, 3/4-n/4, -n/4}, {1/2, 1/2-n/2, -n/2}, 16], {n, 0, 38}] (* Jean-François Alcover, Oct 04 2012 *)
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CROSSREFS
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First differences of A024490.
Sequence in context: A117276 A035295 A006999 * A023430 A023429 A023428
Adjacent sequences: A005249 A005250 A005251 * A005253 A005254 A005255
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from (and formula corrected by) James A. Sellers, Feb 06 2000
Definition revised by N. J. A. Sloane, Aug 16 2009 at the suggestion of Alessandro Orlandi.
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STATUS
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approved
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