A157919 - OEIS

%I #25 Mar 07 2023 02:27:25

%S 49,199,449,799,1249,1799,2449,3199,4049,4999,6049,7199,8449,9799,

%T 11249,12799,14449,16199,18049,19999,22049,24199,26449,28799,31249,

%U 33799,36449,39199,42049,44999,48049,51199,54449,57799,61249,64799,68449,72199,76049,79999

%N a(n) = 50*n^2 - 1.

%C The identity (50*n^2 - 1)^2 - (625*n^2 - 25)*(2*n)^2 = 1 can be written as a(n)^2 - A157918(n)*A005843(n)^2 = 1. - _Vincenzo Librandi_, Feb 10 2012

%H Vincenzo Librandi, <a href="/A157919/b157919.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 From _Vincenzo Librandi_, Feb 10 2012: (Start)

%F G.f.: -x*(49 + 52*x - x^2)/(x-1)^3.

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

%F From _Amiram Eldar_, Mar 07 2023: (Start)

%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(5*sqrt(2)))*Pi/(5*sqrt(2)))/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(5*sqrt(2)))*Pi/(5*sqrt(2)) - 1)/2. (End)

%t LinearRecurrence[{3, -3, 1}, {49, 199, 449}, 50] (* _Vincenzo Librandi_, Feb 10 2012 *)

%t 50 Range[40]^2-1 (* _Harvey P. Dale_, Dec 15 2018 *)

%o (Magma) I:=[49, 199, 449]; [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 10 2012

%o (PARI) for(n=1, 40, print1(50*n^2 - 1", ")); \\ _Vincenzo Librandi_, Feb 10 2012

%Y Cf. A157918, A005843.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 09 2009