A098655 Trace sequence of 3 X 3 symmetric Krawtchouk matrix.

3, 2, 20, 8, 144, 32, 1088, 128, 8448, 512, 66560, 2048, 528384, 8192, 4210688, 32768, 33619968, 131072, 268697600, 524288, 2148532224, 2097152, 17184063488, 8388608, 137455730688, 33554432, 1099578736640, 134217728, 8796361457664

Offset: 0

Author

Paul Barry, Sep 19 2004

Keywords

• easy
• nonn

Comments

Let A=[1,2,1;2,0,-2;1,-2,1] the 3 X 3 symmetric Krawtchouk matrix. Then a(n) = trace(A^n).

Links

• P. Feinsilver and J. Kocik, Krawtchouk Polynomials and Krawtchouk Matrices, Contemporary Mathematics, 287 2001, pp. 83-96.
• Philip Feinsilver, Jerzy Kocik, Krawtchouk matrices from classical and quantum random walks, arXiv:quant-ph/0702173, 2007.
• Index entries for linear recurrences with constant coefficients, signature (2,8,-16).

Crossrefs

Cf. A098656, A098657.

Formula

G.f.: (3 - 4*x - 8*x^2)/((1-2*x)*(1-8*x^2)).

a(n) = 2^n + (2*sqrt(2))^n + (-2*sqrt(2))^n.

a(n) = 2*a(n-1) + 8*a(n-2) - 16*a(n-3).

E.g.f.: exp(2*x) + 2*cosh(2*sqrt(2)*x). - Stefano Spezia, Sep 08 2019