A158598 a(n) = 40*n^2 - 1.

39, 159, 359, 639, 999, 1439, 1959, 2559, 3239, 3999, 4839, 5759, 6759, 7839, 8999, 10239, 11559, 12959, 14439, 15999, 17639, 19359, 21159, 23039, 24999, 27039, 29159, 31359, 33639, 35999, 38439, 40959, 43559, 46239, 48999, 51839, 54759, 57759, 60839, 63999, 67239

Offset

1,1

Comments

The identity (40*n^2 - 1)^2 - (400*n^2 - 20)*(2*n)^2 = 1 can be written as a(n)^2 - A158597(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*(-39 - 42*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 16 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {39, 159, 359}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
40*Range[40]^2-1 (* Harvey P. Dale, Jan 31 2022 *)

Prog

(Magma) I:=[39, 159, 359]; [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 16 2012
(PARI) for(n=1, 40, print1(40*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158597.

Sequence in context: A128826 A240902 A158593 * A105838 A251335 A251328

Adjacent sequences: A158595 A158596 A158597 * A158599 A158600 A158601

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved