A158058 - OEIS

%I #45 Nov 09 2022 01:41:27

%S 14,60,138,248,390,564,770,1008,1278,1580,1914,2280,2678,3108,3570,

%T 4064,4590,5148,5738,6360,7014,7700,8418,9168,9950,10764,11610,12488,

%U 13398,14340,15314,16320,17358,18428,19530,20664,21830,23028,24258,25520

%N a(n) = 16*n^2 - 2*n.

%C The identity (16*(n-1) + 15)^2 - (16*n^2 - 2*n)*4^2 = 1 can be written as A125169(n-1)^2 - a(n)*4^2 = 1. - _Vincenzo Librandi_, Feb 01 2012

%C Sequence found by reading the line from 14, in the direction 14, 60, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012

%C The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 2, 1, 8n-2}]. - _Magus K. Chu_, Nov 08 2022

%H Vincenzo Librandi, <a href="/A158058/b158058.txt">Table of n, a(n) for n = 1..10000</a>

%H E. J. Barbeau, <a href="http://www.math.toronto.edu/barbeau/home.html">Polynomial Excursions</a>, Chapter 10: <a href="http://www.math.toronto.edu/barbeau/hxpol10.pdf">Diophantine equations</a> (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(4^2*t-2)).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: x*(-14 - 18*x)/(x-1)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%p seq(16*n^2-2*n,n=1..40); # _Nathaniel Johnston_, Jun 26 2011

%t LinearRecurrence[{3,-3,1},{14,60,138},40]

%o (Magma) [16*n^2-2*n: n in [1..40]]

%o (PARI) a(n) = 16*n^2-2*n.

%Y Cf. A125169.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 12 2009