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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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..66.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

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

Adjacent sequences:  A106252 A106253 A106254 * A106256 A106257 A106258

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

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Last modified June 30 16:05 EDT 2022. Contains 354943 sequences. (Running on oeis4.)