A202804 - OEIS

%I #54 Dec 12 2024 14:41:06

%S 0,10,32,66,112,170,240,322,416,522,640,770,912,1066,1232,1410,1600,

%T 1802,2016,2242,2480,2730,2992,3266,3552,3850,4160,4482,4816,5162,

%U 5520,5890,6272,6666,7072,7490,7920,8362,8816,9282,9760,10250,10752,11266,11792,12330

%N a(n) = n*(6*n+4).

%C Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033579 in the same spiral. - _Omar E. Pol_, Jul 17 2012

%C Partial sums give A163815. - _Leo Tavares_, Feb 25 2022

%H Jeremy Gardiner, <a href="/A202804/b202804.txt">Table of n, a(n) for n = 0..3000</a>

%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H Leo Tavares, <a href="/A202804/a202804.jpg">Illustration: Star Halves</a>.

%H Pavlos Vavolas, <a href="http://www.dcs.shef.ac.uk/intranet/teaching/projects/archive/msc2005/pdf/m4pv.pdf">Cellular automaton model of cardiac arrhythmias</a>, University of Sheffield Department of Computer Science (2005), (see page 43). [broken link, <a href="http://www.dcs.shef.ac.uk/intranet/archive/public/2004_2005/projects/msc/m4pv.html">abstract</a>]

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

%F a(n) = 2*n(3*n+2) = 6*n^2 + 4*n = 2*A045944(n).

%F a(n) = A080859(n) - 1. - _Omar E. Pol_, Jul 18 2012

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Dec 28 2015

%F G.f.: 2*x*(5 + x)/(1 - x)^3. - _Indranil Ghosh_, Apr 10 2017

%F a(n) = A003154(n+1) - A005408(n). - _Leo Tavares_, Feb 25 2022

%F From _Amiram Eldar_, Mar 01 2022: (Start)

%F Sum_{n>=1} 1/a(n) = (Pi/sqrt(3) - 3*log(3) + 3)/8.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - 3/8. (End)

%F E.g.f.: 2*exp(x)*x*(5 + 3*x). - _Elmo R. Oliveira_, Dec 12 2024

%p A202804:=n->n*(6*n+4): seq(A202804(n), n=0..100); # _Wesley Ivan Hurt_, Apr 09 2017

%t Table[n(6n+4),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,10,32},50] (* _Harvey P. Dale_, Dec 28 2015 *)

%o (PARI) x='x + O('x^50); concat([0], Vec(-2*x*(5 + x)/(x - 1)^3)) \\ _Indranil Ghosh_, Apr 10 2017

%Y Cf. A003154, A005408, A033580, A033581, A049453, A163815, A195319.

%Y Cf. A001318, A033579, A045944, A080859.

%K nonn,easy

%O 0,2

%A _Jeremy Gardiner_, Dec 24 2011