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A106255 Triangle composed of triangular numbers, row sums = A006918.

%I A106255 #31 Feb 14 2022 03:54:03
%S A106255 1,1,1,1,3,1,1,3,3,1,1,3,6,3,1,1,3,6,6,3,1,1,3,6,10,6,3,1,1,3,6,10,10,
%T A106255 6,3,1,1,3,6,10,15,10,6,3,1,1,3,6,10,15,15,10,6,3,1,1,3,6,10,15,21,15,
%U A106255 10,6,3,1
%N A106255 Triangle composed of triangular numbers, row sums = A006918.
%C A106255 Perform the operation Q * R; Q = infinite lower triangular matrix with 1, 2, 3, ... in each column (offset, fill in spaces with zeros). Q = upper right triangular matrix of the form:
%C A106255   1, 1, 1, 1, ...
%C A106255   0, 1, 1, 1, ...
%C A106255   0, 0, 1, 1, ...
%C A106255   0, 0, 0, 1, ...
%C A106255 Q * R generates an array:
%C A106255   1, 1, 1,  1, ...
%C A106255   1, 3, 3,  3, ...
%C A106255   1, 3, 6,  6, ...
%C A106255   1, 3, 6, 10, ...
%C A106255   ...
%C A106255 ... from which we take antidiagonals forming the rows of this triangle.
%H A106255 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A106255 From _Boris Putievskiy_, Jan 13 2013: (Start)
%F A106255 T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals.
%F A106255 a(n) = min(A002260(n)*(A002260(n)+1)/2, A004737(n)*(A004737(n)+1)/2).
%F A106255 a(n) = min(i*(i+1)/2, j*(j+1)/2), where
%F A106255 i = n-t*(t+1)/2,
%F A106255 j = (t*t+3*t+4)/2-n,
%F A106255 t = floor((-1+sqrt(8*n-7))/2). (End)
%e A106255 From _Boris Putievskiy_, Jan 13 2013: (Start)
%e A106255 The start of the sequence as table:
%e A106255   1, 1, 1,  1,  1,  1, ...
%e A106255   1, 3, 3,  3,  3,  3, ...
%e A106255   1, 3, 6,  6,  6,  6, ...
%e A106255   1, 3, 6, 10, 10, 10, ...
%e A106255   1, 3, 6, 10, 15, 15, ...
%e A106255   1, 3, 6, 10, 15, 21, ...
%e A106255   1, 3, 6, 10, 15, 21, ...
%e A106255   ...
%e A106255 (End)
%e A106255 Triangle rows or columns can be generated by following the triangle format:
%e A106255   1;
%e A106255   1, 1;
%e A106255   1, 3, 1;
%e A106255   1, 3, 3,  1;
%e A106255   1, 3, 6,  3,  1;
%e A106255   1, 3, 6,  6,  3,  1;
%e A106255   1, 3, 6, 10,  6,  3,  1;
%e A106255   1, 3, 6, 10, 10,  6,  3,  1;
%e A106255   1, 3, 6, 10, 15, 10,  6,  3, 1;
%e A106255   1, 3, 6, 10, 15, 15, 10,  6, 3, 1;
%e A106255   1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1;
%e A106255   ...
%t A106255 p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*(i + 1), -(2*((n + 1) - i))]], {i, 0, n}]/(2*(1 - x));
%t A106255 Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
%t A106255 Flatten[%]
%Y A106255 Cf. A006918 (row sums, without the zero), A002260, A004736.
%K A106255 nonn,tabl
%O A106255 1,5
%A A106255 _Gary W. Adamson_, Apr 28 2005
%E A106255 Additional comments from _Roger L. Bagula_ and _Gary W. Adamson_, Apr 02 2009