|
%I #29 Aug 26 2019 14:35:33
%S 0,4,36,96,184,300,444,616,816,1044,1300,1584,1896,2236,2604,3000,
%T 3424,3876,4356,4864,5400,5964,6556,7176,7824,8500,9204,9936,10696,
%U 11484,12300,13144,14016,14916,15844,16800,17784,18796,19836,20904,22000,23124
%N 4 times 9-gonal numbers: a(n) = 2*n*(7*n-5).
%C Sequence found by reading the line from 0, in the direction 0, 4, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. The square spiral is related to the primitive Pythagorean triple [3, 4, 5]. - _Omar E. Pol_, Oct 13 2011
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 14n^2 - 10n = A001106(n)*4 = A139268(n)*2.
%F a(n) = a(n-1) + 28*n - 24 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010
%F From _Colin Barker_, Apr 09 2012: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: 4*x*(1+6*x)/(1-x)^3. (End)
%t s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,8!,28}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)
%t 4*PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,4,36},50] (* _Harvey P. Dale_, Aug 26 2019 *)
%o (PARI) a(n)=2*n*(7*n-5) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A001106, A139268, A152759.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, Dec 14 2008
|
