OFFSET
0,3
COMMENTS
Form the 6 node graph with matrix A = [1,1,1,1,0,0; 1,1,0,0,1,1; 1,0,0,0,0,0; 1,0,0,0,0,0; 0,1,0,0,0,0; 0,1,0,0,0,0]. Then this sequence counts walks of length n between the degree 5 vertices.
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,4,-4,-4).
FORMULA
G.f.: x/((1-2*x^2)(1-2*x-2*x^2)).
a(n) = (3+sqrt(3))*(1+sqrt(3))^n/12 + (3-sqrt(3))*(1-sqrt(3))^n/12 - 2^((n-4)/2)*(1+(-1)^n).
a(n) = A002605(n)/2 - 2^((n-4)/2)*(1+(-1)^n).
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n-k+1, k-1)*2^(n-k). - Paul Barry, Oct 23 2004
E.g.f.: exp(x) * (3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) / 6 - cosh(sqrt(2)*x) / 2. - Amiram Eldar, Feb 23 2026
MATHEMATICA
LinearRecurrence[{2, 4, -4, -4}, {0, 1, 2, 8}, 30] (* Harvey P. Dale, Feb 12 2023 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 02 2004
STATUS
approved
