A158588 - OEIS

%I #25 Mar 14 2023 03:36:37

%S 33,135,305,543,849,1223,1665,2175,2753,3399,4113,4895,5745,6663,7649,

%T 8703,9825,11015,12273,13599,14993,16455,17985,19583,21249,22983,

%U 24785,26655,28593,30599,32673,34815,37025,39303,41649,44063,46545,49095,51713,54399,57153

%N a(n) = 34*n^2 - 1.

%C The identity (34*n^2 - 1)^2 - (289*n^2 - 17)*(2*n)^2 = 1 can be written as a(n)^2 - A158587(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158588/b158588.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&amp;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.: x*(-33 - 36*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(34))*Pi/sqrt(34))/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/2. (End)

%t LinearRecurrence[{3, -3, 1}, {33, 135, 305}, 50] (* _Vincenzo Librandi_, Feb 16 2012 *)

%t 34*Range[40]^2-1 (* _Harvey P. Dale_, Feb 10 2015 *)

%o (Magma) I:=[33, 135, 305]; [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 16 2012

%o (PARI) for(n=1, 40, print1(34*n^2-1", ")); \\ _Vincenzo Librandi_, Feb 16 2012

%Y Cf. A005843, A158587.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 22 2009

%E Comment rewritten by _R. J. Mathar_, Oct 16 2009