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

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

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) is the 1,1 element of A^n.

References

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

Links

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

Crossrefs

Cf. A081337, A098655, A098656.

Formula

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