A099163 Expansion of (1-2*x^2)/((1-2*x)*(1+x-x^2)).

1, 1, 2, 3, 7, 12, 27, 49, 106, 199, 419, 804, 1663, 3237, 6618, 13003, 26383, 52156, 105299, 209001, 420586, 836991, 1680747, 3350548, 6718807, 13408957, 26864282, 53653539, 107428471, 214660524, 429638859, 858763489, 1718359018, 3435371767

Offset: 0

Author

Paul Barry, Oct 01 2004

Keywords

• nonn
• easy

Comments

Counts closed walks of length n at the vertex with loop of the graph with adjacency matrix A=[0,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Binomial transform is A099164.

Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ (.5'i + .5i' + .5'ii' + .5e)*(.5j' + .5'kk' + .5'ki' + .5e) ], 1vesforseq = A000079(n+2) (Dement)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3323
• Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
• Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
• Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2).

Crossrefs

Cf. A039834.

Programs

Mathematica

CoefficientList[Series[(1 - 2 x^2)/((1 - 2 x) (1 + x - x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Sep 14 2022 *)

PARI

a(n)=sum(k=-n\5,n\5,binomial(n,(n-5*k)\2)) \\ Paul D. Hanna, Jan 02 2009

PARI

a(n)=-fibonacci(n+1)+2*sum(k=-n\10,n\10,binomial(n,n\2-5*k)) \\ Paul D. Hanna, Jan 02 2009

Formula

a(n)=a(n-1)+3a(n-2)-2a(n-3); a(n)=((sqrt(5)-1)/2)^n(3/10+sqrt(5)/10)+((-sqrt(5)-1)/2)^n(3/10-sqrt(5)/10)+2^(n+1)/5; a(n)=sum{k=0..n, (-1)^(n-k)Fib(n-k+1)(2^(k-1)+0^k/2-sum{j=0..k, C(k, j)j(-1)^j})}.

4*a(n+1) - a(n+3) = A039834(n) - Creighton Dement, Feb 25 2005

Contribution from Paul D. Hanna, Jan 02 2009: (Start)

a(n) = Sum_{k=-[n/5]..[n/5]} C(n, [(n-5*k)/2]).

a(n) = 2*Sum_{k=-[n/10]..[n/10]} C(n, [n/2]-5*k) - fibonacci(n+1). (End)

5*a(n) = 2^(n+1) + A061084(n+1), n>0. - R. J. Mathar, Sep 11 2019