A099177 a(n)=2a(n-1)+4a(n-2)-4a(n-3)-4a(n-4).

0, 1, 2, 8, 20, 60, 160, 448, 1216, 3344, 9120, 24960, 68160, 186304, 508928, 1390592, 3799040, 10379520, 28357120, 77473792, 211661824, 578272256, 1579868160, 4316282880, 11792302080, 32217174016, 88018952192, 240472260608

Offset: 0

Author

Paul Barry, Oct 02 2004

Keywords

• easy
• nonn

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 A099177 counts walks of length n between the degree 5 vertices.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,4,-4,-4).

Crossrefs

Cf. A099176.

Programs

Mathematica

LinearRecurrence[{2,4,-4,-4},{0,1,2,8},30] (* Harvey P. Dale, Feb 12 2023 *)

Formula

G.f.: x/((1-2x^2)(1-2x-2x^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