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A158563 a(n) = 32*n^2 - 1.

%I A158563 #52 Jan 25 2025 23:00:43
%S A158563 31,127,287,511,799,1151,1567,2047,2591,3199,3871,4607,5407,6271,7199,
%T A158563 8191,9247,10367,11551,12799,14111,15487,16927,18431,19999,21631,
%U A158563 23327,25087,26911,28799,30751,32767,34847,36991,39199,41471,43807,46207,48671,51199,53791
%N A158563 a(n) = 32*n^2 - 1.
%C A158563 The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as a(n)^2 - A158562(n)*A005843(n)^2 = 1. [comment rewritten by _R. J. Mathar_, Oct 16 2009]
%C A158563 From _Omar E. Pol_, Apr 21 2021: (Start)
%C A158563 Sequence found by reading the line from 31, in the direction 31, 127, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
%C A158563 The spiral begins as follows:
%C A158563        46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
%C A158563         |                                                        |
%C A158563         |                            0                           |
%C A158563         |                            |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
%C A158563         |                            1                           15
%C A158563         |
%C A158563        51
%C A158563 (End)
%H A158563 Vincenzo Librandi, <a href="/A158563/b158563.txt">Table of n, a(n) for n = 1..1000</a>
%H A158563 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 A158563 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158563 G.f.: x*(-31-34*x+x^2)/(x-1)^3.
%F A158563 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158563 a(n) = A244082(n) - 1. - _Omar E. Pol_, Apr 21 2021
%F A158563 From _Amiram Eldar_, Mar 09 2023: (Start)
%F A158563 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
%F A158563 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/2. (End)
%F A158563 E.g.f.: 1 + exp(x)*(32*x^2 + 32*x - 1). - _Elmo R. Oliveira_, Jan 25 2025
%t A158563 32 Range[40]^2 - 1  (* _Harvey P. Dale_, Mar 04 2011 *)
%t A158563 CoefficientList[Series[(- 31 - 34 x + x^2) / (x - 1)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 11 2013 *)
%o A158563 (Magma) [32*n^2-1: n in [1..40]]; // _Vincenzo Librandi_, Sep 11 2013
%o A158563 (PARI) a(n)=32*n^2-1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A158563 Cf. A005843, A010021, A158562, A158575, A244082.
%Y A158563 Cf. A274979 (generalized 18-gonal numbers).
%K A158563 nonn,easy
%O A158563 1,1
%A A158563 _Vincenzo Librandi_, Mar 21 2009