A171608 Triangle by columns, T(n,k); (..., n, (n+1)) preceded by (n-1) zeros, as an infinite lower triangular matrix.

1, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Dec 12 2009

Keywords

• nonn
• tabl
• easy

Comments

Let the triangle = M as an infinite lower triangular matrix.

M * (1, 2, 3, ...) = A002620: (1, 2, 4, 6, 9, 12, 16, 20, ...);

M * (1, 3, 5, ...) = A084265: (1, 2, 6, 9, 15, 20, 28, 35, ...);

M * (1, 3, 6, ...) = A028724: (1, 2, 6, 9, 18, 24, 40, 50, ...);

Limit_{n->infinity} M^n = A171609: (1, 2, 4, 6, 12, 16, 24, 30, ...).

Examples

			First few rows of the triangle:
  1;
  2, 0;
  0, 2, 0;
  0, 3, 0, 0;
  0, 0, 3, 0, 0;
  0, 0, 4, 0, 0, 0;
  0, 0, 0, 4, 0, 0, 0;
  0, 0, 0, 5, 0, 0, 0, 0;
  0, 0, 0, 0, 5, 0, 0, 0, 0;
  0, 0, 0, 0, 6, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0;
  ...
		

Links

• Micah Manary, Table of n, a(n) for n = 1..5050

Crossrefs

Cf. A002620, A084265, A028724, A171608.

Programs

Maple

A171609 := proc(n,k)
    if k  = ceil(n/2) then
        floor( (n+2)/2) ;
    else
        0;
    end if;
end proc:
seq(seq( A171609(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Sep 23 2021

Formula

Triangle by columns, T(n,k); (..., n, (n+1)) preceded by (n-1) zeros, as an infinite lower triangular matrix.

Extensions

More terms from Micah Manary, Aug 07 2022