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A172045 a(n) = (9*n^4+10*n^3-3*n^2-4*n)/12.

%I A172045 #30 Sep 08 2022 08:45:50
%S A172045 0,1,17,80,240,565,1141,2072,3480,5505,8305,12056,16952,23205,31045,
%T A172045 40720,52496,66657,83505,103360,126560,153461,184437,219880,260200,
%U A172045 305825,357201,414792,479080,550565,629765,717216,813472,919105,1034705
%N A172045 a(n) = (9*n^4+10*n^3-3*n^2-4*n)/12.
%C A172045 The sequence is related to A002414 (octagonal pyramidal numbers) by a(n) = n*A002414(n)-sum(A002414(i), i=1..n-1) for n>0.
%C A172045 This is the case d=3 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6)-sum(k*(k+1)*(2*d*k-2*d+3)/6, k=0..n-1) = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - _Bruno Berselli_, Nov 03 2010
%C A172045 Also, the sequence is related to A000567 by a(n) = sum( i*A000567(i), i=0..n ). [_Bruno Berselli_, Dec 19 2013]
%H A172045 Vincenzo Librandi, <a href="/A172045/b172045.txt">Table of n, a(n) for n = 0..1000</a>
%H A172045 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 A172045 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A172045 a(n) = n*(n+1)*(9*n^2+n-4)/12. - _Bruno Berselli_, Apr 21 2010
%F A172045 G.f. -x*(1 +12*x +5*x^2) / (x - 1)^5 . - _R. J. Mathar_, Nov 17 2011
%t A172045 CoefficientList[Series[x (1 + 12 x + 5 x^2)/(1 - x)^5,{x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 01 2014 *)
%t A172045 LinearRecurrence[{5,-10,10,-5,1},{0,1,17,80,240},40] (* _Harvey P. Dale_, Aug 25 2019 *)
%o A172045 (Magma) [(9*n^4+10*n^3-3*n^2-4*n)/12: n in [0..50]]; // _Vincenzo Librandi_, Jan 01 2014
%Y A172045 Cf. A000567, A002414.
%K A172045 nonn,easy
%O A172045 0,3
%A A172045 _Vincenzo Librandi_, Jan 24 2010
%E A172045 Edited by _Bruno Berselli_, Oct 06 - 12 2010