A133818 a(n) = (8*n+3)*(8*n+5)*(8*n+7)*(8*n+9).

945, 36465, 229425, 801009, 2070705, 4456305, 8473905, 14737905, 23961009, 36954225, 54626865, 77986545, 108139185, 146289009, 193738545, 251888625, 322238385, 406385265, 506025009, 622951665, 759057585, 916333425, 1096868145

Offset: 0

Author

Miklos Kristof, Jan 06 2008, Sep 15 2008

Keywords

• nonn
• easy

Comments

Also 1/3-1/5-1/7+1/9+1/11-1/13-1/15+1/17+1/19--++... = Pi*sqrt(2)/4-1 - Miklos Kristof, Sep 15 2008

Also sum(2*(-1)^n/((4*n+3)*(4*n+5)), n=0..infinity) = Pi*sqrt(2)/4-1 - Miklos Kristof, Sep 15 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Maple

seq((8*n+3)*(8*n+5)*(8*n+7)*(8*n+9), n=0..30);
sum(32*(4*n+3)/((8*n+3)*(8*n+5)*(8*n+7)*(8*n+9)), n=0..infinity) = Pi*sqrt(2)/4-1. Maple: evalf(Pi*sqrt(2)/4-1, 30); gives 0.11072073453959156175397024752... - Miklos Kristof, Sep 15 2008

Mathematica

Times@@@(#+{3,5,7,9}&/@(8Range[0,25])) (* Harvey P. Dale, Mar 14 2011 *)

Formula

G.f.: 3*(315 + 10580*x + 18850*x^2 + 3028*x^3 - 5*x^4)/(1-x)^5.

E.g.f: (945 + 35520*x + 78720*x^2 + 36864*x^3 + 4096*x^4)*exp(x).

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 26 2021