A096886 Expansion of (1+3*x)/(1-8*x^2).

1, 3, 8, 24, 64, 192, 512, 1536, 4096, 12288, 32768, 98304, 262144, 786432, 2097152, 6291456, 16777216, 50331648, 134217728, 402653184, 1073741824, 3221225472, 8589934592, 25769803776, 68719476736, 206158430208, 549755813888

Offset: 0

Author

Paul Barry, Jul 14 2004

Keywords

• easy
• nonn

Comments

From R. J. Mathar, Oct 12 2010: (Start)

Apparently the row n=3 of an array T(n,k) counting walks with k steps on an n X n board starting at an edge position next to a corner, each step to one of the <= 4 adjacent squares:

1, 3, 8, 24, 64, 192, 512, 1536, 4096, 12288, 32768, 98304, 262144, ...

1, 3, 9, 29, 93, 301, 973, 3149, 10189, 32973, 106701, 345293, 1117389, ...

1, 3, 9, 30, 99, 342, 1161, 4050, 13851, 48438, 165969, 580770, 1990899, ...

1, 3, 9, 30, 100, 349, 1216, 4329, 15381, 55187, 197714, 711458, 2557699, ...

1, 3, 9, 30, 100, 350, 1224, 4400, 15776, 57552, 209088, 768768, 2812160, ...

1, 3, 9, 30, 100, 350, 1225, 4409, 15865, 58091, 212586, 786708, 2909166, ...

1, 3, 9, 30, 100, 350, 1225, 4410, 15875, 58200, 213300, 791700, 2936375, ...

1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58211, 213431, 792623, 2943291, ...

1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58212, 213443, 792778, 2944460, ...

1, 3, 9, 30, 100, 350, 1225, 4410, 15876, 58212, 213444, 792791, 2944641, ...

...

(End)

Links

• Index entries for linear recurrences with constant coefficients, signature (0,8).

Crossrefs

Cf. A038754.

Programs

Mathematica

CoefficientList[Series[(1+3x)/(1-8x^2),{x,0,30}],x] (* or *) LinearRecurrence[{0,8},{1,3},30] (* Harvey P. Dale, Apr 25 2023 *)

Formula

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

a(n) = (1 + (-1)^n)*8^floor((n+1)/2)/2 + 3*(1-(-1)^n)*8^floor(n/2)/2.

a(n) = 2^(3*n/2)*(3*sqrt(2)/8 + 1/2 - (3*sqrt(2)/8 - 1/2)*(-1)^n).

a(2n+1) = 2*a(2n) + 2*a(2n-1) + 2*a(2n-2).

a(2n) = 2*a(2n-1) + 2*a(2n-2).

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011