A158557 - OEIS

%I #33 Jan 15 2025 17:36:46

%S 15,240,915,2040,3615,5640,8115,11040,14415,18240,22515,27240,32415,

%T 38040,44115,50640,57615,65040,72915,81240,90015,99240,108915,119040,

%U 129615,140640,152115,164040,176415,189240,202515,216240,230415,245040,260115,275640,291615

%N a(n) = 225*n^2 + 15.

%C The identity (30*n^2 + 1)^2 - (225*n^2 + 15)*(2*n)^2 = 1 can be written as A158558(n)^2 - a(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158557/b158557.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: 15*(1 + 13*x + 16*x^2)/(1-x)^3.

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

%F From _Amiram Eldar_, Mar 09 2023: (Start)

%F Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(15))*Pi/sqrt(15) + 1)/30.

%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(15))*Pi/sqrt(15) + 1)/30. (End)

%F E.g.f.: 15*exp(x)*(1 + 15*x + 15*x^2). - _Elmo R. Oliveira_, Jan 15 2025

%t LinearRecurrence[{3, -3, 1}, {15, 240, 915}, 50] (* _Vincenzo Librandi_, Feb 14 2012 *)

%t 225*Range[0,40]^2+15 (* _Harvey P. Dale_, Apr 06 2019 *)

%o (Magma) I:=[15, 240, 915]; [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 14 2012

%o (PARI) for(n=0, 22, print1(225*n^2 + 15", ")); \\ _Vincenzo Librandi_, Feb 14 2012

%Y Cf. A005843, A158558.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Mar 21 2009

%E Comment rewritten, a(0) added by _R. J. Mathar_, Oct 16 2009