A086570 Expansion of (1 + 3x + 5x^2 + 7x^3 + ...) / (1 - 2x + 3x^2 - 4x^3 + ...).

1, 5, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428

Offset: 0

Author

Gary W. Adamson, Jul 22 2003

Keywords

• nonn
• easy

Comments

Row sums of number triangle A113128. - Paul Barry, Oct 14 2005

The Engel expansion of 1 + exp(1/8)*sqrt(2*Pi)*erf(1/(2*sqrt(2)))/5 = 1.2175306077808... - Benedict W. J. Irwin, Dec 16 2016

Examples

			a(6) = 44 = 8 + a(5) = 8 + 36.
		

Links

• Leo Tavares, Square illustration
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A078370, A016754.

Programs

Mathematica

CoefficientList[Series[(z^3 + 3*z^2 + 3*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* and *) Join[{1, 5}, Table[4*(2*(n + 1) + 1), {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

PARI

a(n)=if(n>1, 8*n-4, 4*n+1) \\ Charles R Greathouse IV, Dec 16 2016

Formula

a(0) = 1, a(1) = 5, a(2) = 12; then a(n+1) = a(n) + 8, n > 2.

From Paul Barry, Oct 14 2005: (Start)

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

a(n) = 8n - 4 + 4*C(0, n) + C(1, n);

a(n) = C(n+1, n) + 3*C(n, n-1) + 3*C(n-1, n-2) + C(n-2, n-3). (End)

a(n) = A017113(n-1), n > 1. - R. J. Mathar, Sep 12 2008