A158657 a(n) = 784*n^2 - 28.

756, 3108, 7028, 12516, 19572, 28196, 38388, 50148, 63476, 78372, 94836, 112868, 132468, 153636, 176372, 200676, 226548, 253988, 282996, 313572, 345716, 379428, 414708, 451556, 489972, 529956, 571508, 614628, 659316, 705572, 753396, 802788, 853748, 906276, 960372

Offset: 1

Author

Vincenzo Librandi, Mar 23 2009

Keywords

• nonn
• easy

Comments

The identity (56*n^2 - 1)^2 - (784*n^2 - 28)*(2*n)^2 = 1 can be written as A158658(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, A158554, A158658.

Programs

Magma

I:=[756, 3108, 7028]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 17 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {756, 3108, 7028}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

PARI

for(n=1, 40, print1(784*n^2 - 28", ")); \\ Vincenzo Librandi, Feb 17 2012

Formula

G.f.: 28*x*(-27 - 30*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 20 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)))/56.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)) - 1)/56. (End)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: 28*(exp(x)*(28*x^2 + 28*x - 1) + 1).

a(n) = 28*A158554(n). (End)

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009