A114960 Expansion of (-1+3*x-5*x^2+4*x^3) / ((1-2*x)*(2*x^2-1)*(x-1)^2).

1, 1, 6, 11, 30, 57, 128, 247, 518, 1013, 2068, 4083, 8242, 16369, 32880, 65519, 131310, 262125, 524780, 1048555, 2098154, 4194281, 8390632, 16777191, 33558502, 67108837, 134225892, 268435427, 536887266, 1073741793

Offset: 1

Author

Creighton Dement, Feb 21 2006

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-6,10,-4).

Crossrefs

Cf. A113979.

Programs

Mathematica

CoefficientList[Series[(-1+3x-5x^2+4x^3)/((1-2x)(2x^2-1)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[ {4,-3,-6,10,-4},{1,1,6,11,30},40] (* Harvey P. Dale, Aug 11 2023 *)

PARI

Vec(x*(1 - 3*x + 5*x^2 - 4*x^3) / ((1 - x)^2*(1 - 2*x)*(1 - 2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

Formula

a(n) = 2^n - n - 1 - ((-1)^n - 1)*2^(1/2*n)*sqrt(2)/4.

a(n+2) - 2*a(n+1) + a(n) = A113979(n+4) - A113979(n+3).

a(n) = 4*a(n-1) - 3*a(n-2) - 6*a(n-3) + 10*a(n-4) - 4*a(n-5) for n>5. - Colin Barker, May 18 2019