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%I #15 Jun 17 2017 03:53:50
%S 1,14,90,360,1075,2646,5684,11040,19845,33550,53966,83304,124215,
%T 179830,253800,350336,474249,630990,826690,1068200,1363131,1719894,
%U 2147740,2656800,3258125,3963726,4786614,5740840,6841535,8104950
%N First differences of triangular numbers with square pyramidal indices.
%C This sequence is also the sums of a partition of the integers into groups of (n+1)^2 integers starting at 1 and not repeating or skipping any: a(0)=1, a(1)=2+3+4+5=14, a(2)=6+7+8+9+10+11+12+13+14=90, etc.
%H Harvey P. Dale, <a href="/A077538/b077538.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F Let SP(m) be the m-th square pyramidal number m*(m+1)*(2*m+1)/6 and let T(k) be the k-th Triangular number k*(k+1)/2; then a(n) = T(SP(n+1))-T(SP(n)) = ((n+1)^2*(n+2)*(2*n^2+2*n+3))/6.
%F G.f.: (1+8*x+21*x^2+10*x^3)/(1-x)^6. [_Colin Barker_, Apr 30 2012]
%e SP(3)=14, SP(4)=30, T(14)=105 and T(30)=465, so a(3)=465-105=360.
%t nn=30;Join[{1},With[{tr=Accumulate[Range[(nn(nn+1)(2nn+1))/6]]},Differences[ Table[tr[[n]],{n,Accumulate[Range[nn]^2]}]]]](* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,14,90,360,1075,2646},30] (* _Harvey P. Dale_, Mar 07 2013 *)
%K nonn,easy
%O 0,2
%A _Amarnath Murthy_, Nov 09 2002
%E More terms and better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 14 2002
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