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Triangle composed of triangular numbers, row sums = A006918.
4

%I #31 Feb 14 2022 03:54:03

%S 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 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 10,6,3,1

%N Triangle composed of triangular numbers, row sums = A006918.

%C 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 1, 1, 1, 1, ...

%C 0, 1, 1, 1, ...

%C 0, 0, 1, 1, ...

%C 0, 0, 0, 1, ...

%C Q * R generates an array:

%C 1, 1, 1, 1, ...

%C 1, 3, 3, 3, ...

%C 1, 3, 6, 6, ...

%C 1, 3, 6, 10, ...

%C ...

%C ... from which we take antidiagonals forming the rows of this triangle.

%H 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 From _Boris Putievskiy_, Jan 13 2013: (Start)

%F T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals.

%F a(n) = min(A002260(n)*(A002260(n)+1)/2, A004737(n)*(A004737(n)+1)/2).

%F a(n) = min(i*(i+1)/2, j*(j+1)/2), where

%F i = n-t*(t+1)/2,

%F j = (t*t+3*t+4)/2-n,

%F t = floor((-1+sqrt(8*n-7))/2). (End)

%e From _Boris Putievskiy_, Jan 13 2013: (Start)

%e The start of the sequence as table:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 3, 3, 3, 3, 3, ...

%e 1, 3, 6, 6, 6, 6, ...

%e 1, 3, 6, 10, 10, 10, ...

%e 1, 3, 6, 10, 15, 15, ...

%e 1, 3, 6, 10, 15, 21, ...

%e 1, 3, 6, 10, 15, 21, ...

%e ...

%e (End)

%e Triangle rows or columns can be generated by following the triangle format:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 3, 3, 1;

%e 1, 3, 6, 3, 1;

%e 1, 3, 6, 6, 3, 1;

%e 1, 3, 6, 10, 6, 3, 1;

%e 1, 3, 6, 10, 10, 6, 3, 1;

%e 1, 3, 6, 10, 15, 10, 6, 3, 1;

%e 1, 3, 6, 10, 15, 15, 10, 6, 3, 1;

%e 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1;

%e ...

%t 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 Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];

%t Flatten[%]

%Y Cf. A006918 (row sums, without the zero), A002260, A004736.

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, Apr 28 2005

%E Additional comments from _Roger L. Bagula_ and _Gary W. Adamson_, Apr 02 2009