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A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).

%I A152751 #47 Dec 27 2024 18:30:42
%S A152751 0,3,24,63,120,195,288,399,528,675,840,1023,1224,1443,1680,1935,2208,
%T A152751 2499,2808,3135,3480,3843,4224,4623,5040,5475,5928,6399,6888,7395,
%U A152751 7920,8463,9024,9603,10200,10815,11448,12099,12768,13455,14160,14883,15624,16383,17160
%N A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).
%C A152751 a(n) also can be represented as n concentric triangles (see example). - _Omar E. Pol_, Aug 21 2011
%H A152751 Ivan Panchenko, <a href="/A152751/b152751.txt">Table of n, a(n) for n = 0..1000</a>
%H A152751 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A152751 a(n) = 9*n^2 - 6*n = 3*A000567(n) = A064201(n)/3.
%F A152751 a(n) = a(n-1) + 18*n - 15 with n > 0, a(0)=0. - _Vincenzo Librandi_, Nov 26 2010
%F A152751 G.f.: 3*x*(1+5*x)/(1-x)^3. - _Bruno Berselli_, Jan 21 2011
%F A152751 From _Elmo R. Oliveira_, Dec 25 2024: (Start)
%F A152751 E.g.f.: 3*exp(x)*x*(1 + 3*x).
%F A152751 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
%F A152751 a(n) = n + A152995(n). (End)
%e A152751 From _Omar E. Pol_, Aug 21 2011: (Start)
%e A152751 Illustration of initial terms as concentric triangles:
%e A152751 .
%e A152751 .                                          o
%e A152751 .                                         o o
%e A152751 .                                        o   o
%e A152751 .                                       o     o
%e A152751 .                 o                    o   o   o
%e A152751 .                o o                  o   o o   o
%e A152751 .               o   o                o   o   o   o
%e A152751 .              o     o              o   o     o   o
%e A152751 .    o        o   o   o            o   o   o   o   o
%e A152751 .   o o      o   o o   o          o   o   o o   o   o
%e A152751 .           o           o        o   o           o   o
%e A152751 .          o o o o o o o o      o   o o o o o o o o   o
%e A152751 .                              o                       o
%e A152751 .                             o o o o o o o o o o o o o o
%e A152751 .
%e A152751 .    3            24                       63
%e A152751 (End)
%t A152751 s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,18}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)
%t A152751 3*PolygonalNumber[8,Range[0,40]] (* _Harvey P. Dale_, May 08 2022 *)
%o A152751 (PARI) a(n)=3*n*(3*n-2) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A152751 Cf. A000567, A064201, A139267, A152995.
%Y A152751 3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152759, A152767, A153783, A153448, A153875.
%Y A152751 Cf. A033581, A085250, A152734, A194273. - _Omar E. Pol_, Aug 21 2011
%Y A152751 Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=18: see Comments lines of A226492.
%K A152751 easy,nonn
%O A152751 0,2
%A A152751 _Omar E. Pol_, Dec 12 2008