OFFSET
0,3
COMMENTS
Counts closed walks of length n at a vertex of a triangle, to which a loop has been added at one of the other vertices.
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [0, 1, 1; 1, 0, 1; 1, 1, 1].
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7.
Index entries for linear recurrences with constant coefficients, signature (1,3,1).
FORMULA
a(n) = ((1+sqrt(2))^n + (1-sqrt(2))^n + 2*(-1)^n)/4.
a(n) = a(n-1) + 3*a(n-2) + a(n-3).
a(n) = (1/2)*((-1)^n + Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k).
a(n) = ((-1)^n + A001333(n))/2.
E.g.f.: (cosh(x) + exp(x)*cosh(sqrt(2)*x) - sinh(x))/2. - Stefano Spezia, Mar 31 2024
MATHEMATICA
LinearRecurrence[{1, 3, 1}, {1, 0, 2}, 41] (* or *) Table[(LucasL[n, 2] +2*(-1)^n)/4, {n, 0, 40}] (* G. C. Greubel, Aug 18 2022 *)
PROG
(PARI) Vec((1-x-x^2)/(1-x-3*x^2-x^3) + O(x^50)) \\ Michel Marcus, Mar 25 2014
(Magma) [(Evaluate(DicksonFirst(n, -1), 2) +2*(-1)^n)/4: n in [0..40]]; // G. C. Greubel, Aug 18 2022
(SageMath) [(lucas_number2(n, 2, -1) +2*(-1)^n)/4 for n in (0..40)] # G. C. Greubel, Aug 18 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 22 2004
STATUS
approved