cp's OEIS Frontend

A158443 a(n) = 16*n^2 - 4.

%I A158443 #31 Jan 16 2025 07:57:22
%S A158443 12,60,140,252,396,572,780,1020,1292,1596,1932,2300,2700,3132,3596,
%T A158443 4092,4620,5180,5772,6396,7052,7740,8460,9212,9996,10812,11660,12540,
%U A158443 13452,14396,15372,16380,17420,18492,19596,20732,21900,23100,24332,25596,26892,28220,29580
%N A158443 a(n) = 16*n^2 - 4.
%C A158443 The identity (8*n^2 - 1)^2 - (16*n^2 - 4) *(2*n)^2 = 1 can be written as A157914(n)^2 - a(n)*A005843(n)^2 = 1.
%C A158443 Sequence found by reading the line from 12, in the direction 12, 60, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012
%H A158443 Vincenzo Librandi, <a href="/A158443/b158443.txt">Table of n, a(n) for n = 1..10000</a>
%H A158443 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 A158443 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158443 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158443 G.f: 4*x*(3+6*x-x^2)/(1-x)^3.
%F A158443 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158443 Sum_{n>=1} 1/a(n) = 1/8.
%F A158443 Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/16. (End)
%F A158443 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158443 E.g.f.: 4*(exp(x)*(4*x^2 + 4*x - 1) + 1).
%F A158443 a(n) = 4*A000466(n). (End)
%t A158443 16Range[60]^2-4  (* _Harvey P. Dale_, Mar 18 2011 *)
%o A158443 (Magma) I:=[12, 60, 140]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
%o A158443 (PARI) a(n) = 16*n^2 - 4.
%Y A158443 Cf. A000466, A005843, A074377, A157914.
%K A158443 nonn,easy
%O A158443 1,1
%A A158443 _Vincenzo Librandi_, Mar 19 2009