A158771 a(n) = 78*n^2 - 1.

77, 311, 701, 1247, 1949, 2807, 3821, 4991, 6317, 7799, 9437, 11231, 13181, 15287, 17549, 19967, 22541, 25271, 28157, 31199, 34397, 37751, 41261, 44927, 48749, 52727, 56861, 61151, 65597, 70199, 74957, 79871, 84941, 90167, 95549, 101087, 106781, 112631, 118637

Offset

1,1

Comments

The identity (78*n^2 - 1)^2 - (1521*n^2 - 39)*(2*n)^2 = 1 can be written as a(n)^2 - A158770(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).

Formula

G.f.: x*(-77 - 80*x + 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>=1} 1/a(n) = (1 - cot(Pi/sqrt(78))*Pi/sqrt(78))/2.

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

Mathematica

LinearRecurrence[{3, -3, 1}, {77, 311, 701}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
78*Range[40]^2-1 (* Harvey P. Dale, Nov 28 2018 *)

Prog

(Magma) I:=[77, 311, 701]; [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=1, 40, print1(78*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158770.

Sequence in context: A156652 A298102 A158767 * A331976 A020206 A020304

Adjacent sequences: A158768 A158769 A158770 * A158772 A158773 A158774

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved