A153794 - OEIS

%I #30 Dec 04 2022 18:28:15

%S 0,4,32,84,160,260,384,532,704,900,1120,1364,1632,1924,2240,2580,2944,

%T 3332,3744,4180,4640,5124,5632,6164,6720,7300,7904,8532,9184,9860,

%U 10560,11284,12032,12804,13600,14420,15264,16132,17024

%N 4 times octagonal numbers: a(n) = 4*n*(3*n-2).

%C Sequence found by reading the segment (0, 4) together with the line from 4, in the direction 4, 32, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 18 2012

%H G. C. Greubel, <a href="/A153794/b153794.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = 12*n^2 - 8*n = 4*A000567(n) = 2*A139267(n).

%F a(n) = 24*n + a(n-1) - 20 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010

%F a(0)=0, a(1)=4, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jul 14 2011

%F G.f.: 4*(x + 5*x^2)/(1-x)^3. - _Harvey P. Dale_, Jul 14 2011

%F E.g.f.: 4*x*(1 + 3*x)*exp(x). - _G. C. Greubel_, Aug 29 2016

%t s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,7!,24}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)

%t Table[4n(3n-2),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,4,32},41] (* _Harvey P. Dale_, Jul 14 2011 *)

%t 4*PolygonalNumber[8,Range[0,40]] (* _Harvey P. Dale_, Dec 04 2022 *)

%o (PARI) a(n) = 12*n^2 - 8*n; \\ _Altug Alkan_, Aug 29 2016

%Y Cf. A000567, A139267, A152751, A153808.

%K easy,nonn

%O 0,2

%A _Omar E. Pol_, Jan 19 2009