cp's OEIS Frontend

A158575 a(n) = 32*n^2 + 1.

%I A158575 #51 Jan 16 2025 20:56:22
%S A158575 1,33,129,289,513,801,1153,1569,2049,2593,3201,3873,4609,5409,6273,
%T A158575 7201,8193,9249,10369,11553,12801,14113,15489,16929,18433,20001,21633,
%U A158575 23329,25089,26913,28801,30753,32769,34849,36993,39201,41473,43809,46209,48673,51201,53793
%N A158575 a(n) = 32*n^2 + 1.
%C A158575 The identity (32*n^2 + 1)^2 - (256*n^2 + 16)*(2*n)^2 = 1 can be written as a(n)^2-A158574(n)*A005843(n)^2 = 1. - Comment rewritten by _R. J. Mathar_, Oct 16 2009
%C A158575 Sequence found by reading the line segment from 1 to 33 together with the line from 33, in the direction 33, 129, ..., in the square spiral whose vertices are the generalized 18-gonal numbers A274979. - _Omar E. Pol_, Apr 21 2021
%H A158575 Vincenzo Librandi, <a href="/A158575/b158575.txt">Table of n, a(n) for n = 0..10000</a>
%H A158575 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 A158575 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158575 G.f.: (1+30*x+33*x^2)/(1-x)^3.
%F A158575 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158575 For n > 0 a(n) = sqrt(8*(A000217(4*n-1)^2 + A000217(4*n)^2) + 1). - _J. M. Bergot_, Sep 03 2015
%F A158575 a(n) = A244082(n) + 1. - _Omar E. Pol_, Apr 21 2021
%F A158575 From _Amiram Eldar_, Mar 09 2023: (Start)
%F A158575 Sum_{n>=0} 1/a(n) = (1 + coth(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
%F A158575 Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2. (End)
%F A158575 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158575 E.g.f.: exp(x)*(1 + 32*x + 32*x^2).
%F A158575 a(n) = A081585(2*n). (End)
%t A158575 LinearRecurrence[{3, -3, 1}, {1, 33, 129}, 50] (* _Vincenzo Librandi_, Feb 15 2012 *)
%t A158575 32*Range[0,40]^2+1 (* _Harvey P. Dale_, Jul 20 2021 *)
%o A158575 (Magma) I:=[1, 33, 129]; [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 A158575 (PARI) for(n=0, 50, print1(32*n^2+1", ")); \\ _Vincenzo Librandi_, Feb 15 2012
%Y A158575 Cf. A000217, A005843, A010021, A081585, A158563, A158574, A244082.
%Y A158575 Cf. A274979 (generalized 18-gonal numbers).
%K A158575 nonn,easy
%O A158575 0,2
%A A158575 _Vincenzo Librandi_, Mar 21 2009
%E A158575 a(0) added by _R. J. Mathar_, Oct 16 2009