A005252 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k,2k).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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) 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

a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an even number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 2; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, two of which have an even number of ones. See the E. Munarini et al. reference, p. 323. - Emeric Deutsch, Jun 28 2015

a(n) is the number of even permutations p of 1,2,...,n such that |p(i)-i| <= 1 for i=1,2,...,n. - Dmitry Efimov, Jan 08 2016

This sequence (prefixed with 0) is an autosequence of the first kind, whose second kind companion is (2 followed by abs(A111734)). - Jean-François Alcover, Oct 30 2017

a(n+1) is the number of n-bit sequences such that 1's appear in groups of three or more. Example: for n = 5, a(6) = 7 because we have 00000, 00111, 01110, 11100, 11110, 01111, 11111. Source: exercise 1.11 in I. Stewart. - João Camarneiro, Dec 23 2024

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.
• David J. C. MacKay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 251.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Ian Stewart, Galois theory, CRC Press, Boca Raton, FL, 2015, p. 32.
• E. L. Tan, On the cycle graph of a graph and inverse cycle graphs, Ph.D. Dissertation, Univ. of Philippines, Diliman, Quezon City, 1987.
• E. L. Tan, On Fibonacci numbers and cycle graphs, Matimyas Matemaka (Published by the Mathematical Society of the Philippines), 13 (No. 2, 1990), 1-4.

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.
• R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
• V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,2).
• V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
• 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
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
• S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim. 25 (2013), 505-522, DOI:10.1007/s10878-011-9433-z.
• E. Munarini and N. Z. Salvi, Structural and enumerative properties of the Fibonacci cubes, Discrete Math., 255, 2002, 317-324.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• E. L. Tan, Letter to N. J. A. Sloane, Feb 1992
• OEIS Wiki, Autosequence.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Crossrefs

First differences of A024490.

Cf. A005251, A005676, A007318, A010892.

Cf. A106511, A111734, A173021.

Programs

Haskell

a005252 n = sum $ map (\x -> a007318 (n - x) x) [0, 2 .. 2 * div n 4]
-- Reinhard Zumkeller, Jul 05 2013

Magma

I:=[1,1,1,1]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

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 *)

PARI

Vec((1-x)/((1-x-x^2)*(1-x+x^2)) + O(x^100)) \\ Altug Alkan, Jan 08 2015

PARI

a(n) = fibonacci(n+1)>>1 + (n%6<2); \\ Kevin Ryde, Apr 29 2021

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.

From Paul Barry, Mar 15 2004: (Start)

a(n) = Fibonacci(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. (End)

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

a(n) = A173021(2^(n-1) - 1) for n > 0. - Reinhard Zumkeller, Feb 07 2010

Limit_{n->oo} 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

a(n) = Fibonacci(n+1) + (-1)^(n+1)*A106511(n+2). - Katharine Ahrens, May 05 2019

E.g.f.: exp(x/2)*(15*(cos(sqrt(3)*x/2) + cosh(sqrt(5)*x/2)) + 5*sqrt(3)*sin(sqrt(3)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - Stefano Spezia, Aug 03 2022

Extensions

More terms from (and formula corrected by) James Sellers, Feb 06 2000

Definition revised at the suggestion of Alessandro Orlandi by N. J. A. Sloane, Aug 16 2009