A158562 - OEIS

%I #30 Oct 05 2024 22:16:00

%S 240,1008,2288,4080,6384,9200,12528,16368,20720,25584,30960,36848,

%T 43248,50160,57584,65520,73968,82928,92400,102384,112880,123888,

%U 135408,147440,159984,173040,186608,200688,215280,230384,246000,262128,278768,295920,313584,331760

%N a(n) = 256*n^2 - 16.

%C The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as A158563(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by _R. J. Mathar_, Oct 16 2009]

%H Vincenzo Librandi, <a href="/A158562/b158562.txt">Table of n, a(n) for n = 1..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.: 16*x*(-15 - 18*x + x^2)/(x-1)^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>=1} 1/a(n) = (4 - Pi)/128.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 4)/128. (End)

%t 16(16Range[40]^2-1) (* or *) LinearRecurrence[{3,-3,1},{240,1008,2288},40] (* _Harvey P. Dale_, Sep 13 2011 *)

%o (Magma) I:=[240,1008,2288]; [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 15 2012

%o (PARI) for(n=1, 50, print1(256*n^2-16", ")); \\ _Vincenzo Librandi_, Feb 15 2012

%Y Cf. A005843, A158563.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 21 2009