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%I #21 Aug 18 2022 08:46:28
%S 1,1,1,4,4,4,10,10,10,20,20,20,35,35,35,56,56,56,84,84,84,120,120,120,
%T 165,165,165,220,220,220,286,286,286,364,364,364,455,455,455,560,560,
%U 560,680,680,680,816,816,816,969,969,969
%N Triplicated tetrahedral numbers A000292.
%C The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).
%C The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3,0,-3,3,0,1,-1).
%F a(n) = binomial(floor(n/3)+3,3).
%F a(n) + a(n-1) + a(n-2) = A144677(n).
%F a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).
%F G.f.: 1/((x-1)^4*(x^2+x+1)^3).
%F Sum_{n>=0} 1/a(n) = 9/2. - _Amiram Eldar_, Aug 18 2022
%p A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);
%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* _Harvey P. Dale_, Mar 09 2018 *)
%Y Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.
%K nonn,easy
%O 0,4
%A _Johannes W. Meijer_, May 18 2011
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