A158560 a(n) = 30*n^2 - 1.

29, 119, 269, 479, 749, 1079, 1469, 1919, 2429, 2999, 3629, 4319, 5069, 5879, 6749, 7679, 8669, 9719, 10829, 11999, 13229, 14519, 15869, 17279, 18749, 20279, 21869, 23519, 25229, 26999, 28829, 30719, 32669, 34679, 36749, 38879, 41069, 43319, 45629, 47999, 50429

Offset: 1

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (30*n^2-1)^2 - (225*n^2-15) * (2*n)^2 = 1 can be written as a(n)^2 - A158559(n) * A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158559.

Programs

Magma

I:=[29, 119, 269]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 14 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {29, 119, 269}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
30*Range[50]^2-1 (* Harvey P. Dale, May 30 2018 *)

PARI

for(n=1, 40, print1(30*n^2-1", ")); \\ Vincenzo Librandi, Feb 14 2012

Formula

G.f.: x*(-29 - 32*x + x^2)/(x - 1)^3.

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

From Amiram Eldar, Mar 09 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(30))*Pi/sqrt(30))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(30))*Pi/sqrt(30) - 1)/2. (End)

Extensions

Comment rewritten by R. J. Mathar, Oct 16 2009