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%I #38 Apr 01 2021 01:43:35
%S 0,3,14,38,80,145,238,364,528,735,990,1298,1664,2093,2590,3160,3808,
%T 4539,5358,6270,7280,8393,9614,10948,12400,13975,15678,17514,19488,
%U 21605,23870,26288,28864,31603,34510,37590,40848,44289,47918,51740,55760
%N a(n) = n*(n+1)*(5*n + 4)/6.
%C Partial sums of A005475.
%C Suppose we extend the triangle in A215631 to a symmetric array by reflection about the main diagonal. The array is defined by m(i,j) = i^2 + i*j + j^2: 3, 7, 13, ...; 7, 12, 19, ...; 13, 19, 27, .... Then a(n) is the sum of the n-th antidiagonal. Examples: 3, 7 + 7, 13 + 12 + 13, 21 + 19 + 19 + 21, etc. - _J. M. Bergot_, Jun 25 2013
%C Binomial transform of [0,3,8,5,0,0,0,...]. - _Alois P. Heinz_, Mar 10 2015
%H Jinyuan Wang, <a href="/A162147/b162147.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F From _R. J. Mathar_, Jun 27 2009: (Start)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)
%F a(n) = A033994(n) + A000217(n).
%F G.f.: x*(3+2*x)/(1-x)^4. (End)
%F a(n) = A035005(n+1)/4. - _Johannes W. Meijer_, Feb 04 2010
%F a(n) = Sum_{i=0..n} i*(n + 1 + i). - _Bruno Berselli_, Mar 17 2016
%F E.g.f.: x*(18 + 24*x + 5*x^2)*exp(x)/6. - _G. C. Greubel_, Apr 01 2021
%e For n=4, a(4) = 0*(5+0) + 1*(5+1) + 2*(5+2) + 3*(5+3) + 4*(5+4) = 80. - _Bruno Berselli_, Mar 17 2016
%p A162147:= n-> n*(n+1)*(5*n+4)/6; seq(A162147(n), n=0..40); # _G. C. Greubel_, Apr 01 2021
%t Table[(n(n+1)(5n+4))/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,14,38},50] (* _Harvey P. Dale_, May 04 2013 *)
%o (PARI) a(n)=n*(n+1)*(5*n+4)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [n*(n+1)*(5*n+4)/6: n in [0..40]]; // _G. C. Greubel_, Apr 01 2021
%o (Sage) [n*(n+1)*(5*n+4)/6 for n in (0..40)] # _G. C. Greubel_, Apr 01 2021
%Y Cf. A002412, A002413, A006331, A016061, A033994.
%K nonn,easy
%O 0,2
%A _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009
%E Definition rephrased by _R. J. Mathar_, Jun 27 2009
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