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 A099163 Expansion of (1-2x^2)/((1-2x)(1+x-x^2));. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. LINKS Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750. Index entries for linear recurrences with constant coefficients, signature (1,3,-2). 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) PROG Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ (.5'i + .5i' + .5'ii' + .5e)*(.5j' + .5'kk' + .5'ki' + .5e) ], 1vesforseq = A000079(n+2) (Dement) Contribution from Paul D. Hanna, Jan 02 2009: (Start) (PARI) a(n)=sum(k=-n\5, n\5, binomial(n, (n-5*k)\2)) (PARI) a(n)=-fibonacci(n+1)+2*sum(k=-n\10, n\10, binomial(n, n\2-5*k)) (End) CROSSREFS Cf. A039834. Sequence in context: A054272 A259593 A129016 * A000676 A283823 A263658 Adjacent sequences:  A099160 A099161 A099162 * A099164 A099165 A099166 KEYWORD easy,nonn AUTHOR Paul Barry, Oct 01 2004 STATUS approved

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