%I #42 Sep 08 2022 08:45:41

%S 72,136,200,264,328,392,456,520,584,648,712,776,840,904,968,1032,1096,

%T 1160,1224,1288,1352,1416,1480,1544,1608,1672,1736,1800,1864,1928,

%U 1992,2056,2120,2184,2248,2312,2376,2440,2504,2568,2632,2696,2760,2824,2888,2952

%N a(n) = 8*(8*n + 1).

%C The identity (128*n^2+32*n+1)^2-(4*n^2+n)*(64*n+8)^2=1 can be written as A157337(n)^2-A007742(n)*a(n)^2=1. This is the case s=2 of the identity (8*n^2*s^4+8*n*s^2+1)^2-(n^2*s^2+n)*(8*n*s^3+4*s)^2=1. - _Vincenzo Librandi_, Jan 29 2012

%C Likewise, the immediate identity (a(n)^2+1)^2-(a(n)^2+2)*a(n)^2 = 1 can be rewritten as A158686(8*n+1)^2-(A158686(8*n+1)+1)*a(n)^2=1. - _Bruno Berselli_, Feb 13 2012

%H Vincenzo Librandi, <a href="/A157336/b157336.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F From _Vincenzo Librandi_, Jan 29 2012: (Start)

%F G.f.: 8*(1+7*x)/(x-1)^2. [corrected by _Georg Fischer_, May 12 2019]

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

%F E.g.f.: 8*(1+8*x)*exp(x). - _G. C. Greubel_, Feb 01 2018

%t Range[72, 5000, 64] (* _Vladimir Joseph Stephan Orlovsky_, Jul 16 2011 *)

%t LinearRecurrence[{2, -1}, {72, 136}, 50] (* _Vincenzo Librandi_, Jan 29 2012 *)

%o (Magma) I:=[72, 136]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Jan 29 2012

%o (PARI) for(n=1, 40, print1(64*n + 8", ")); \\ _Vincenzo Librandi_, Jan 29 2012

%Y Cf. A007742, A157337.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Feb 27 2009