Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #24 Mar 14 2023 03:36:40
%S 272,1139,2584,4607,7208,10387,14144,18479,23392,28883,34952,41599,
%T 48824,56627,65008,73967,83504,93619,104312,115583,127432,139859,
%U 152864,166447,180608,195347,210664,226559,243032,260083,277712,295919,314704,334067,354008,374527
%N a(n) = 289*n^2 - 17.
%C The identity (34*n^2 - 1)^2 - (289*n^2 - 17) * (2*n)^2 = 1 can be written as A158588(n)^2 - a(n) * A005843(n)^2 = 1.
%H Vincenzo Librandi, <a href="/A158587/b158587.txt">Table of n, a(n) for n = 1..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 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: 17*x*(-16 - 19*x + x^2)/(x-1)^3.
%F From _Amiram Eldar_, Mar 14 2023: (Start)
%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(17))*Pi/sqrt(17))/34.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(17))*Pi/sqrt(17) - 1)/34. (End)
%t LinearRecurrence[{3, -3, 1}, {272, 1139, 2584}, 50] (* _Vincenzo Librandi_, Feb 15 2012 *)
%t 289*Range[40]^2-17 (* _Harvey P. Dale_, Jan 30 2019 *)
%o (Magma) I:=[272, 1139, 2584]; [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(289*n^2-17", ")); \\ _Vincenzo Librandi_, Feb 15 2012
%Y Cf. A005843, A158588.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 22 2009
%E Comment rewritten by _R. J. Mathar_, Oct 16 2009