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A085788 Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2) = t(3)+t(6) = 6+21 = 27.

%I A085788 #37 Aug 14 2025 15:15:26
%S A085788 6,27,72,150,270,441,672,972,1350,1815,2376,3042,3822,4725,5760,6936,
%T A085788 8262,9747,11400,13230,15246,17457,19872,22500,25350,28431,31752,
%U A085788 35322,39150,43245,47616,52272,57222,62475,68040,73926,80142,86697,93600,100860,108486
%N A085788 Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2) = t(3)+t(6) = 6+21 = 27.
%H A085788 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A085788 a(n) = (3/2)*n*(n+1)^2 = 3*A006002(n).
%F A085788 a(n) = Sum_{j=1..n} (j+n+1)*(n+1). - _Zerinvary Lajos_, Sep 10 2006
%F A085788 From _Colin Barker_, Mar 17 2014: (Start)
%F A085788 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A085788 G.f.: 3*x*(x+2)/(x-1)^4. (End)
%F A085788 E.g.f.: 3*exp(x)*x*(1 + x)*(4 + x)/2. - _Elmo R. Oliveira_, Aug 14 2025
%p A085788 a:=n->sum(sum(sum(j-k+1, j=1..n), k=0..n),m=0..n): seq(a(n), n=1..45); # _Zerinvary Lajos_, May 30 2007
%t A085788 LinearRecurrence[{4,-6,4,-1},{6,27,72,150},50] (* _Harvey P. Dale_, Dec 14 2017 *)
%o A085788 (PARI) v=vector(40,i,i*(i+1)/2); s=0; forstep(i=3,40,3,s+=v[i]; print1(s","))
%Y A085788 Cf. A004188, A006002.
%Y A085788 Row sums of triangle A001283.
%Y A085788 Cf. A254407. - _Bruno Berselli_, Jan 30 2015
%K A085788 nonn,easy
%O A085788 1,1
%A A085788 _Jon Perry_, Jul 23 2003
%E A085788 Edited and more terms from _Michel Marcus_, Mar 17 2014