A158484 a(n) = 49*n^2 - 7.

42, 189, 434, 777, 1218, 1757, 2394, 3129, 3962, 4893, 5922, 7049, 8274, 9597, 11018, 12537, 14154, 15869, 17682, 19593, 21602, 23709, 25914, 28217, 30618, 33117, 35714, 38409, 41202, 44093, 47082, 50169, 53354, 56637, 60018, 63497, 67074, 70749, 74522, 78393

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 A158485(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, A158485.

Programs

Magma

I:=[42, 189, 434]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

Mathematica

LinearRecurrence[{3,-3,1},{42,189,434},50]

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*(-6-9*x+x^2)/(x-1)^3.

From Amiram Eldar, Mar 05 2023: (Start)

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

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

E.g.f.: 7*(exp(x)*(7*x^2 + 7*x - 1) + 1). - Elmo R. Oliveira, Jan 27 2025