A158628 a(n) = 44*n^2 - 1.

43, 175, 395, 703, 1099, 1583, 2155, 2815, 3563, 4399, 5323, 6335, 7435, 8623, 9899, 11263, 12715, 14255, 15883, 17599, 19403, 21295, 23275, 25343, 27499, 29743, 32075, 34495, 37003, 39599, 42283, 45055, 47915, 50863, 53899, 57023, 60235, 63535, 66923, 70399

Offset

1,1

Comments

The identity (44*n^2 - 1)^2 - (484*n^2 - 22)*(2*n)^2 = 1 can be written as a(n)^2 - A158627(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*(-43 - 46*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(11)))*Pi/(2*sqrt(11)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(11)))*Pi/(2*sqrt(11)) - 1)/2. (End)

Mathematica

44Range[0, 40]^2-1 (* or *) CoefficientList[Series[(1-46 x-43 x^2)/ (x-1)^3, {x, 0, 40}], x] (* Harvey P. Dale, Apr 22 2011 *)
LinearRecurrence[{3, -3, 1}, {43, 175, 395}, 40] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[43, 175, 395]; [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(44*n^2-1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158627.

Sequence in context: A057816 A162295 A187722 * A123597 A138631 A309905

Adjacent sequences: A158625 A158626 A158627 * A158629 A158630 A158631

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

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

Status

approved