%I #35 Feb 27 2024 12:04:44
%S 10,110,410,910,1610,2510,3610,4910,6410,8110,10010,12110,14410,16910,
%T 19610,22510,25610,28910,32410,36110,40010,44110,48410,52910,57610,
%U 62510,67610,72910,78410,84110,90010,96110,102410,108910,115610,122510,129610,136910,144410
%N a(n) = 100*n^2 + 10.
%C The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as A158493(n)^2 - a(n)*A005843(n)^2 = 1. - _Vincenzo Librandi_, Feb 21 2012
%H Vincenzo Librandi, <a href="/A158492/b158492.txt">Table of n, a(n) for n = 0..10000</a>
%H Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%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 21 2012: (Start)
%F G.f.: -(10 + 80*x + 110*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 05 2023: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + coth(Pi/sqrt(10))*Pi/sqrt(10))/20.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/sqrt(10))*Pi/sqrt(10))/20. (End)
%t LinearRecurrence[{3, -3, 1}, {10, 110, 410}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)
%t 100Range[0,40]^2+10 (* _Harvey P. Dale_, Dec 30 2019 *)
%o (Magma) I:=[10, 110, 410]; [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 21 2012
%o (PARI) for(n=0, 40, print1(100*n^2 + 10", ")); \\ _Vincenzo Librandi_, Feb 21 2012
%Y Cf. A005843, A158493.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, Mar 21 2009
%E Edited by _N. J. A. Sloane_, Oct 12 2009