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%I #32 Mar 12 2024 12:47:18
%S 1,1,2,3,7,12,27,49,106,199,419,804,1663,3237,6618,13003,26383,52156,
%T 105299,209001,420586,836991,1680747,3350548,6718807,13408957,
%U 26864282,53653539,107428471,214660524,429638859,858763489,1718359018,3435371767
%N Expansion of (1-2*x^2)/((1-2*x)*(1+x-x^2)).
%C 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.
%C Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ (.5'i + .5i' + .5'ii' + .5e)*(.5j' + .5'kk' + .5'ki' + .5e) ], 1vesforseq = A000079(n+2) (Dement)
%H Michael De Vlieger, <a href="/A099163/b099163.txt">Table of n, a(n) for n = 0..3323</a>
%H Michael A. Allen, <a href="https://arxiv.org/abs/2209.01377">On a Two-Parameter Family of Generalizations of Pascal's Triangle</a>, arXiv:2209.01377 [math.CO], 2022.
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H Johann Cigler, <a href="https://arxiv.org/abs/2212.02118">Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials</a>, arXiv:2212.02118 [math.NT], 2022.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2).
%F 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})}.
%F 4*a(n+1) - a(n+3) = A039834(n) - _Creighton Dement_, Feb 25 2005
%F Contribution from _Paul D. Hanna_, Jan 02 2009: (Start)
%F a(n) = Sum_{k=-[n/5]..[n/5]} C(n, [(n-5*k)/2]).
%F a(n) = 2*Sum_{k=-[n/10]..[n/10]} C(n, [n/2]-5*k) - fibonacci(n+1). (End)
%F 5*a(n) = 2^(n+1) + A061084(n+1), n>0. - _R. J. Mathar_, Sep 11 2019
%t CoefficientList[Series[(1 - 2 x^2)/((1 - 2 x) (1 + x - x^2)), {x, 0, 33}], x] (* _Michael De Vlieger_, Sep 14 2022 *)
%o (PARI) a(n)=sum(k=-n\5,n\5,binomial(n,(n-5*k)\2)) \\ _Paul D. Hanna_, Jan 02 2009
%o (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
%Y Cf. A039834.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Oct 01 2004
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