A157804 - OEIS

%I #23 Sep 08 2022 08:45:42

%S 4954951,1658764801,6277377151,13860792001,24409009351,37922029201,

%T 54399851551,73842476401,96249903751,121622133601,149959165951,

%U 181261000801,215527638151,252759078001,292955320351,336116365201,382242212551

%N a(n) = 1482401250*n^2 - 2793393900*n + 1315947601.

%C The identity (1482401250*n^2 - 2793393900*n + 1315947601)^2 - (27225*n^2 - 51302*n + 24168)*(8984250*n - 8464830)^2 = 1 can be written as a(n)^2 - A157802(n)*A157803(n)^2 = 1.

%C This is the case s=165 and r=25651 of the identity (2*(s^2*n-r)^2-1)^2 - (((s^2*n-r)^2-1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2-1)/s^2 is an integer if r^2 == 1 (mod s^2). Therefore, for s=165, nonnegative r values are: 1, 1574, 6049, 7624, 19601, 21176, 25651, 27224, ... - _Bruno Berselli_, Apr 24 2018

%H Vincenzo Librandi, <a href="/A157804/b157804.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.: x*(4954951 + 1643899948*x + 1315947601*x^2)/(1 - x)^3.

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

%t LinearRecurrence[{3, -3, 1}, {4954951, 1658764801, 6277377151}, 30]

%o (Magma) I:=[4954951, 1658764801, 6277377151]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..30]];

%o (PARI) a(n) = 1482401250*n^2 - 2793393900*n + 1315947601;

%Y Cf. A157802, A157803.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 07 2009