A077866 Expansion of (1-x)^(-1)/(1 - x - 2*x^2 + 2*x^3).

1, 2, 5, 8, 15, 22, 37, 52, 83, 114, 177, 240, 367, 494, 749, 1004, 1515, 2026, 3049, 4072, 6119, 8166, 12261, 16356, 24547, 32738, 49121, 65504, 98271, 131038, 196573, 262108, 393179, 524250, 786393, 1048536, 1572823, 2097110, 3145685, 4194260, 6291411, 8388562

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

Equals triangle A122196 * [1,2,4,8,16,...]. - Gary W. Adamson, Nov 29 2008

Conjecture: let b(n) be the number of subsets S of {1,2,...,n} having more than one element such that (sum of least two elements of S) = max(S). Then b(0) = b(1) = b(2) = 0 and b(n+3) = a(n) for n >= 0. - Clark Kimberling Sep 27 2022

Examples

			G.f. = 1 + 2*x + 5*x^2 + 8*x^3 + 15*x^4 + 22*x^5 + 37*x^6 + ... - _Michael Somos_, Aug 11 2021
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Crossrefs

Bisections are A005803 and A050488.

Cf. A052551 (first differences), A122196.

Programs

Mathematica

CoefficientList[Series[(1-x)^(-1)/(1-x-2x^2+2x^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-4,2},{1,2,5,8},50] (* Harvey P. Dale, Feb 16 2013 *)

PARI

Vec((1-x)^(-1)/(1-x-2*x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 2^(n/2)*(3 + 2*sqrt(2) + (3 - 2*sqrt(2))*(-1)^n) - n - 5. - Paul Barry, Apr 23 2004

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + 2*a(n-4); a(0)=1, a(1)=2, a(2)=5, a(3)=8. - Harvey P. Dale, Feb 16 2013

a(2n) = 3*2^(n+1) - 2(n+1) - 3 = A050488(n+1) and a(2n+1) = 2^(n+3) - 2(n+3) = A005803(n+3). Also, a(2n+1) - a(2n) = 2^(n+1) - 1 = a(2n) - a(2n - 1). - Gregory L. Simay, Feb 07 2021

E.g.f.: 6*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) - exp(x)*(5 + x). - Stefano Spezia, Feb 08 2021

G.f.: 1/((1 - x)^2 * (1 - 2*x^2)). - Michael Somos, Aug 11 2021