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A106255 Triangle composed of triangular numbers, row sums = A006918. 5

%I

%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 Row sums = A006918, (deleting the zero): 1, 2, 5, 8, 14, 20, 30 ,...

%C Row sums are: {1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, ...}.

%C T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals. - _Boris Putievskiy_, Jan 13 2013

%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 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:

%F 1 1 1 1 ...

%F 0 1 1 1 ...

%F 0 0 1 1 ...

%F 0 0 0 1 ...

%F Q * R generates an array:

%F 1 1 1 1 ...

%F 1 3 3 3 ...

%F 1 3 6 6 ...

%F 1 3 6 10 ..

%F ...

%F ... from which we take antidiagonals forming the rows of triangle A106255.

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

%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 ...

%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} (End)

%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, 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

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)