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%I #43 Sep 08 2022 08:45:50
%S 0,1,21,78,190,375,651,1036,1548,2205,3025,4026,5226,6643,8295,10200,
%T 12376,14841,17613,20710,24150,27951,32131,36708,41700,47125,53001,
%U 59346,66178,73515,81375,89776,98736,108273,118405,129150,140526
%N a(n) = n*(n+1)*(6*n-5)/2.
%C Generated by formula: n*(n+1)*(2*d*n-2*d+3)/6 with d=9.
%C This sequence is related to A051682 by a(n) = n*A051682(n) - Sum_{i=0..n-1} A051682(i); in fact this is the case d=9 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n -2*d + 3)/6. - _Bruno Berselli_, Apr 16 2012
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - _Bruno Berselli_, Feb 13 2014
%H Vincenzo Librandi, <a href="/A172082/b172082.txt">Table of n, a(n) for n = 0..1000</a>
%H Bruno 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), 2008.
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(0)=0, a(1)=1, a(2)=21, a(3)=78; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Jun 29 2011
%F G.f.: x*(1+17*x)/(1-x)^4. - _Harvey P. Dale_, Jun 29 2011
%F a(n) = Sum_{i=0..n-1} (n-i)*(18*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F E.g.f.: x*(2 + 19*x + 6*x^2)*exp(x)/2. - _G. C. Greubel_, Aug 30 2019
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2*(3*sqrt(3)*Pi + 9*log(3) + 12*log(2) - 5)/55.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(6*Pi + 6*sqrt(3)*log(sqrt(3)+2) - 16*log(2) + 5)/55. (End)
%p seq(n*(n+1)*(6*n-5)/2, n=0..40); # _G. C. Greubel_, Aug 30 2019
%t Table[(18n^3+3n^2-15n)/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,21,78}, 40] (* _Harvey P. Dale_, Jun 29 2011 *)
%t CoefficientList[Series[x*(1+17*x)/(1-x)^4, {x,0,40}], x] (* _Vincenzo Librandi_, Jan 02 2014 *)
%o (Magma) [(18*n^3+3*n^2-15*n)/6: n in [0..40]]; // _Vincenzo Librandi_, Jan 02 2014
%o (PARI) vector(40, n, n*(n-1)*(6*n-11)/2) \\ _G. C. Greubel_, Aug 30 2019
%o (Sage) [n*(n+1)*(6*n-5)/2 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o (GAP) List([0..40], n-> n*(n+1)*(6*n-5)/2); # _G. C. Greubel_, Aug 30 2019
%Y Cf. A051682.
%Y Cf. similar sequences listed in A237616.
%K nonn,easy
%O 0,3
%A _Vincenzo Librandi_, Jan 25 2010