A153448 - OEIS

%I #52 Feb 02 2025 17:37:18

%S 0,3,36,99,192,315,468,651,864,1107,1380,1683,2016,2379,2772,3195,

%T 3648,4131,4644,5187,5760,6363,6996,7659,8352,9075,9828,10611,11424,

%U 12267,13140,14043,14976,15939,16932,17955,19008,20091,21204

%N 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

%C This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - _Bruno Berselli_, Aug 26 2010

%H G. C. Greubel, <a href="/A153448/b153448.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

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

%F a(n) = 15*n^2 - 12*n = A051624(n)*3.

%F a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F G.f.: 3*x*(1 + 9*x)/(1-x)^3. - _Bruno Berselli_, Jan 21 2011

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - _Harvey P. Dale_, Jun 18 2014

%F E.g.f.: 3*x*(1 + 5*x)*exp(x). - _G. C. Greubel_, Aug 21 2016

%F a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - _Charlie Marion_, Jul 29 2021

%t Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* _Harvey P. Dale_, Jun 18 2014 *)

%t 3*PolygonalNumber[12,Range[0,60]] (* _Harvey P. Dale_, May 13 2022 *)

%o (PARI) a(n)=3*n*(5*n-4) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A051624, A152965.

%Y 3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.

%Y Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Dec 26 2008