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

1,2

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, ...).

Links

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

Formula

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

Example

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;
  ...

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

Crossrefs

Cf. A002620, A084265, A028724, A171608.

Sequence in context: A359240 A280285 A033719 * A307985 A024164 A138805

Adjacent sequences: A171605 A171606 A171607 * A171609 A171610 A171611

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Dec 12 2009

Extensions

More terms from Micah Manary, Aug 07 2022

Status

approved