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A005252 Sum_{k=0..floor(n/4)} binomial(n-2k,2k).
(Formerly M1048)
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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]

a(n) is the number of length n binary words that have an even number of 0's and every 0 is immediately followed by a 1. a(6) = 7 because we have: 010111, 011011, 011101, 101011, 101101, 110101, 111111. - Geoffrey Critzer, Jan 08 2014

REFERENCES

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.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 424

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index to sequences with linear recurrences with constant coefficients, signature (2,-1,0,1).

FORMULA

Second differences give sequence shifted twice - E. L. Tan, Univ. Phillipines.

G.f.: (1-x)/((1-x-x^2)(1-x+x^2)). Simon Plouffe in his 1992 dissertation.

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

lim(n->00,a(n)/a(n+1))=(sqrt(5)-1)/2. - Sergei N. Gladkovskii, Jan 05 2014

G.f.: (1+ Q(0)*x^4/2)/(1-x), where Q(k) = 1 + 1/(1 - x*( 4*k+2 -x +x^3)/( x*( 4*k+4 -x +x^3) +1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 07 2014

MAPLE

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, Mar 26 2008

MATHEMATICA

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] (* 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 *)

PROG

(Haskell)

a005252 n = sum $ map (\x -> a007318 (n - x) x) [0, 2 .. 2 * div n 4]

-- Reinhard Zumkeller, Jul 05 2013

CROSSREFS

First differences of A024490.

Cf. A007318.

Sequence in context: A117276 A035295 A006999 * A023430 A023429 A023428

Adjacent sequences:  A005249 A005250 A005251 * A005253 A005254 A005255

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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.

STATUS

approved

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Last modified July 29 21:54 EDT 2014. Contains 245046 sequences.