A158769 a(n) = 78*n^2 + 1.

1, 79, 313, 703, 1249, 1951, 2809, 3823, 4993, 6319, 7801, 9439, 11233, 13183, 15289, 17551, 19969, 22543, 25273, 28159, 31201, 34399, 37753, 41263, 44929, 48751, 52729, 56863, 61153, 65599, 70201, 74959, 79873, 84943, 90169, 95551, 101089, 106783, 112633, 118639

Offset

0,2

Comments

The identity (78*n^2 + 1)^2 - (1521*n^2 + 39)*(2*n)^2 = 1 can be written as a(n)^2 - A158768(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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).

Formula

G.f.: -(1 + 76*x + 79*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 23 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(78))*Pi/sqrt(78) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(78))*Pi/sqrt(78) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 79, 313}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
78*Range[0, 40]^2+1 (* Harvey P. Dale, Dec 06 2018 *)

Prog

(Magma) I:=[1, 79, 313]; [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 21 2012
(PARI) for(n=0, 40, print1(78*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158768.

Sequence in context: A219478 A082077 A341182 * A158774 A157507 A142897

Adjacent sequences: A158766 A158767 A158768 * A158770 A158771 A158772

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

Comment rewritten, a(0) added, and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved