cp's OEIS Frontend

A157446 a(n) = 16*n^2 - n.

%I A157446 #25 Sep 22 2022 22:29:53
%S A157446 15,62,141,252,395,570,777,1016,1287,1590,1925,2292,2691,3122,3585,
%T A157446 4080,4607,5166,5757,6380,7035,7722,8441,9192,9975,10790,11637,12516,
%U A157446 13427,14370,15345,16352,17391,18462,19565,20700,21867,23066,24297,25560
%N A157446 a(n) = 16*n^2 - n.
%C A157446 The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as A157448(n)^2 - a(n)*A157447(n)^2 = 1. - _Vincenzo Librandi_, Jan 26 2012
%C A157446 This is the case s=4 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - _Bruno Berselli_, Jan 26 2012
%C A157446 Sequence found by reading the line from 15, in the direction 15, 62, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012
%C A157446 The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 6, 1, 8n-2}]. For n=1, this collapses to [3; {1, 6}]. - _Magus K. Chu_, Sep 22 2022
%H A157446 Vincenzo Librandi, <a href="/A157446/b157446.txt">Table of n, a(n) for n = 1..10000</a>
%H A157446 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>
%H A157446 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157446 G.f.: x*(15 + 17*x)/(1-x)^3. - _Vincenzo Librandi_, Jan 26 2012
%F A157446 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jan 26 2012
%t A157446 LinearRecurrence[{3,-3,1},{15,62,141},40] (* _Vincenzo Librandi_, Jan 26 2012 *)
%o A157446 (Magma) I:=[15, 62, 141]; [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_, Jan 26 2012
%o A157446 (PARI) for(n=1, 22, print1(16*n^2 - n", ")); \\ _Vincenzo Librandi_, Jan 26 2012
%Y A157446 Cf. A157447, A157448.
%K A157446 nonn,easy
%O A157446 1,1
%A A157446 _Vincenzo Librandi_, Mar 01 2009