A158481 a(n) = 49*n^2 + 7.

56, 203, 448, 791, 1232, 1771, 2408, 3143, 3976, 4907, 5936, 7063, 8288, 9611, 11032, 12551, 14168, 15883, 17696, 19607, 21616, 23723, 25928, 28231, 30632, 33131, 35728, 38423, 41216, 44107, 47096, 50183, 53368, 56651, 60032, 63511, 67088, 70763, 74536, 78407

Offset: 1

Author

Vincenzo Librandi, Mar 20 2009

Keywords

• nonn
• easy

Comments

The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as A158482(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158482, A247541.

Programs

Magma

I:=[56, 203, 448];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Mathematica

LinearRecurrence[{3,-3,1},{56,203,448},40]

PARI

a(n)=49*n^2+7.

Formula

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

G.f: 7*x*(8+5*x+x^2)/(1-x)^3.

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(7))*Pi/sqrt(7) - 1)/14.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(7))*Pi/sqrt(7))/14. (End)

From Elmo R. Oliveira, Jan 15 2025: (Start)

E.g.f.: 7*(exp(x)*(7*x^2 + 7*x + 1) - 1).

a(n) = 7*A247541(n). (End)