A141384 Trace of the n-th power of a certain 8X8 adjacency matrix.

8, 8, 32, 158, 828, 4408, 23564, 126106, 675076, 3614144, 19349432, 103593806, 554625900, 2969386480, 15897666068, 85113810058, 455687062276, 2439682811480, 13061709929936, 69930511268510, 374397872321628

Offset: 0

Author

Gerard P. Michon, Jun 29 2008

Keywords

• easy
• nonn

Comments

a(n) is the trace of the n-th power of the adjacency matrix of order 8 whose rows are (up to simultaneous permutations of the rows and columns): 10111010 01111001 01111001 10111010 00111000 11111111 11111111 11111111.

For n>2, this is also the number of ways to mark one edge at every vertex of a regular n-gonal prism so that no edge is marked at both extremities.

Remarkably, for n>1, a(n)=A141221(n)+2.

The fourth-order linear recurrence established by Max Alekseyev for A141221, based on the minimal polynomial of the above (singular) matrix, namely x(x-1)(x^3-7x^2+9x-1) = x^5-8*x^4+16*x^3-10*x^2+x. Since its degree is 5, the corresponding recurrence holds for corresponding elements of the successive powers (or sums thereof, including matrix traces) only for n>=5. The recurrence would be valid down to n=4 if we had a(0)=4, which is not the case.

Examples

			a(0) = 8 because the trace of the order-8 identity matrix is 8.
a(1) = 8 because all diagonal elements of the adjacency matrix are 1 (there's a loop at each vertex).
		

Links

• Max A. Alekseyev, Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.
• G. P. Michon, A screaming game for short-sighted people.
• G. P. Michon, Silent circles, enumerated by Max Alekseyev.
• G. P. Michon, Brocoum's Screaming Circles.
• Index entries for linear recurrences with constant coefficients, signature (8,-16,10,-1).

Crossrefs

Cf. A141221.

Formula

For n>=5, a(n) = 8*a(n-1)-16*a(n-2)+10*a(n-3)-a(n-4).

For positive values of n: a(n) = (5.3538557854308)^n + (1.5235479602692)^n + 1 + (0.1225962542999)^n. The dominant term in the above is the n-th power of (7+2*sqrt(22)*cos(atan(sqrt(5319)/73)/3))/3.

G.f.: 2*(4-28*x+48*x^2-25*x^3+2*x^4)/((1-x)*(1-7*x+9*x^2-x^3)). [Colin Barker, Jan 20 2012]

Extensions

Edited by Max Alekseyev, Aug 03 2015