A069071 a(n) = (2*n + 1)*((2*n + 1)^4 + 4).

5, 255, 3145, 16835, 59085, 161095, 371345, 759435, 1419925, 2476175, 4084185, 6436435, 9765725, 14349015, 20511265, 28629275, 39135525, 52522015, 69344105, 90224355, 115856365, 147008615, 184528305, 229345195, 282475445, 345025455, 418195705, 503284595, 601692285

Offset: 0

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn
• changed

Comments

The formula for Pi in the formula section was discovered by the mathematician and astronomer Nilakantha Somayaji (1444-1544) (Roy, 1990). - Amiram Eldar, Jan 18 2023

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A019683.

Programs

Mathematica

a[n_] := (2*n + 1)*((2*n + 1)^4 + 4); Array[a, 30, 0] (* Amiram Eldar, Jul 16 2022 *)

Formula

Pi = 16 * Sum_{n>=0} (-1)^n/a(n).

From Elmo R. Oliveira, Sep 03 2025: (Start)

G.f.: 5*(1 + x)*(1 + 44*x + 294*x^2 + 44*x^3 + x^4)/(x-1)^6.

E.g.f.: (5 + 250*x + 1320*x^2 + 1360*x^3 + 400*x^4 + 32*x^5)*exp(x).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). (End)