cp's OEIS Frontend

A158493 a(n) = 20*n^2 + 1.

%I A158493 #38 Aug 06 2025 10:56:05
%S A158493 1,21,81,181,321,501,721,981,1281,1621,2001,2421,2881,3381,3921,4501,
%T A158493 5121,5781,6481,7221,8001,8821,9681,10581,11521,12501,13521,14581,
%U A158493 15681,16821,18001,19221,20481,21781,23121,24501,25921,27381,28881,30421,32001,33621,35281
%N A158493 a(n) = 20*n^2 + 1.
%C A158493 The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as a(n)^2 - A158492(n)*A005843(n)^2 = 1. - _Vincenzo Librandi_, Feb 21 2012
%C A158493 Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 81, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - _Omar E. Pol_, Nov 05 2012
%H A158493 Vincenzo Librandi, <a href="/A158493/b158493.txt">Table of n, a(n) for n = 0..10000</a>
%H A158493 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 A158493 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158493 From _Vincenzo Librandi_, Feb 21 2012: (Start)
%F A158493 G.f.: -(1 + 18*x + 21*x^2)/(x-1)^3.
%F A158493 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A158493 From _Amiram Eldar_, Mar 06 2023: (Start)
%F A158493 Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2.
%F A158493 Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2. (End)
%F A158493 From _Elmo R. Oliveira_, Jan 25 2025: (Start)
%F A158493 E.g.f.: exp(x)*(1 + 20*x + 20*x^2).
%F A158493 a(n) = A212656(2*n). (End)
%t A158493 LinearRecurrence[{3, -3, 1}, {1, 21, 81}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)
%t A158493 20*Range[0,50]^2+1 (* _Harvey P. Dale_, Aug 06 2025 *)
%o A158493 (Magma) I:=[1, 21, 81]; [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 A158493 (PARI) for(n=0, 40, print1(20*n^2 + 1", ")); \\ _Vincenzo Librandi_, Feb 21 2012
%Y A158493 Cf. A005843, A158492, A195162, A212656.
%K A158493 nonn,easy
%O A158493 0,2
%A A158493 _Vincenzo Librandi_, Mar 21 2009
%E A158493 Edited by _N. J. A. Sloane_, Oct 12 2009