OFFSET
0,2
COMMENTS
Second binomial transform of expansion of (1-3*x)/(1-4*x). Third binomial transform of A054878 (closed walks at a vertex of K_4). With interpolated zeros, counts closed walks of length n at the vertices of the edge-vertex incidence graph of K_4 associated with the vertices of K_4.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12).
FORMULA
a(n) = 2^(n-2)*(3^n + 3) = (6^n + 3*2^n)/4.
G.f.: U(0)/4 where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012
G.f.: U(0)/4 where U(k)= 1 + 3/( 3^k - 2*x*9^k/(2*x*3^k + 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 31 2012
E.g.f.: (1/4)*( 3*exp(2*x) + exp(6*x) ). - G. C. Greubel, Jan 04 2023
MATHEMATICA
LinearRecurrence[{8, -12}, {1, 3}, 41] (* G. C. Greubel, Jan 04 2023 *)
CoefficientList[Series[(1-5x)/((1-2x)(1-6x)), {x, 0, 30}], x] (* Harvey P. Dale, Jan 30 2024 *)
PROG
(Magma) [3*(6^(n-1) + 2^(n-1))/2: n in [0..40]]; // G. C. Greubel, Jan 04 2023
(SageMath) [3*(6^(n-1) +2^(n-1))/2 for n in range(41)] # G. C. Greubel, Jan 04 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 06 2004
STATUS
approved