A158658 a(n) = 56*n^2 - 1.

55, 223, 503, 895, 1399, 2015, 2743, 3583, 4535, 5599, 6775, 8063, 9463, 10975, 12599, 14335, 16183, 18143, 20215, 22399, 24695, 27103, 29623, 32255, 34999, 37855, 40823, 43903, 47095, 50399, 53815, 57343, 60983, 64735, 68599, 72575, 76663, 80863, 85175, 89599

Offset

1,1

Comments

The identity (56*n^2 - 1)^2 - (784*n^2 - 2 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158657(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*(-55 - 58*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(14)))*Pi/(2*sqrt(14)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(14)))*Pi/(2*sqrt(14)) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {55, 223, 503}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
56 Range[40]^2-1 (* Harvey P. Dale, Oct 23 2022 *)

Prog

(Magma) I:=[55, 223, 503]; [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
(PARI) for(n=1, 40, print1(56*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158657.

Sequence in context: A254148 A271738 A280888 * A296036 A286854 A020182

Adjacent sequences: A158655 A158656 A158657 * A158659 A158660 A158661

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

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

Status

approved