A098656 -id:A098656 - cp's OEIS frontend

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

Paul Barry, Sep 19 2004

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

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).

Cf. A098656, A098657.

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

A098657 Expansion of (1-x-4x^2)/((1-2x)(1-8x^2)).

Original entry on oeis.org

1, 1, 6, 4, 40, 16, 288, 64, 2176, 256, 16896, 1024, 133120, 4096, 1056768, 16384, 8421376, 65536, 67239936, 262144, 537395200, 1048576, 4297064448, 4194304, 34368126976, 16777216, 274911461376, 67108864, 2199157473280, 268435456, 17592722915328, 1073741824

Offset: 0

Paul Barry, Sep 19 2004

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

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

Index entries for linear recurrences with constant coefficients, signature (2,8,-16).

Cf. A081337, A098655, A098656.

a(n) = 2^((3*n-4)/2)*(1+(-1)^n)+2^(n-1).
a(n) = 2*a(n-1) + 8*a(n-2) - 16*a(n-3).
a(2n) = A081337(n) = (8^n+4^n)/2 and a(2n+1) = 4^n. - Peter Kagey, Jul 14 2023

cp's OEIS Frontend

A098655 Trace sequence of 3 X 3 symmetric Krawtchouk matrix.

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