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A106255
Triangle composed of triangular numbers, row sums = A006918.
4
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, 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, 10, 6, 3, 1
OFFSET
1,5
COMMENTS
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:
1, 1, 1, 1, ...
0, 1, 1, 1, ...
0, 0, 1, 1, ...
0, 0, 0, 1, ...
Q * R generates an array:
1, 1, 1, 1, ...
1, 3, 3, 3, ...
1, 3, 6, 6, ...
1, 3, 6, 10, ...
...
... from which we take antidiagonals forming the rows of this triangle.
LINKS
FORMULA
From Boris Putievskiy, Jan 13 2013: (Start)
T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals.
a(n) = min(A002260(n)*(A002260(n)+1)/2, A004737(n)*(A004737(n)+1)/2).
a(n) = min(i*(i+1)/2, j*(j+1)/2), where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)
EXAMPLE
From Boris Putievskiy, Jan 13 2013: (Start)
The start of the sequence as table:
1, 1, 1, 1, 1, 1, ...
1, 3, 3, 3, 3, 3, ...
1, 3, 6, 6, 6, 6, ...
1, 3, 6, 10, 10, 10, ...
1, 3, 6, 10, 15, 15, ...
1, 3, 6, 10, 15, 21, ...
1, 3, 6, 10, 15, 21, ...
...
(End)
Triangle rows or columns can be generated by following the triangle format:
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, 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, 10, 6, 3, 1;
...
MATHEMATICA
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));
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]
CROSSREFS
Cf. A006918 (row sums, without the zero), A002260, A004736.
Sequence in context: A046540 A123191 A157454 * A143086 A327481 A152714
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 28 2005
EXTENSIONS
Additional comments from Roger L. Bagula and Gary W. Adamson, Apr 02 2009
STATUS
approved