A158603 a(n) = 441*n^2 + 21.
21, 462, 1785, 3990, 7077, 11046, 15897, 21630, 28245, 35742, 44121, 53382, 63525, 74550, 86457, 99246, 112917, 127470, 142905, 159222, 176421, 194502, 213465, 233310, 254037, 275646, 298137, 321510, 345765, 370902, 396921, 423822, 451605, 480270, 509817, 540246
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..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).
Programs
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Magma
I:=[21, 462, 1785]; [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
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Mathematica
LinearRecurrence[{3, -3, 1}, {21, 462, 1785}, 50] (* Vincenzo Librandi, Feb 16 2012 *) -
PARI
for(n=0, 40, print1(441*n^2 + 21", ")); \\ Vincenzo Librandi, Feb 16 2012
Formula
G.f.: -21*(1 + 19*x + 22*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>=0} 1/a(n) = (coth(Pi/sqrt(21))*Pi/sqrt(21) + 1)/42.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(21))*Pi/sqrt(21) + 1)/42. (End)
E.g.f.: 21*exp(x)*(1 + 21*x + 21*x^2). - Elmo R. Oliveira, Jan 15 2025
Extensions
Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009
Comments