login
Triangle by columns, T(n,k); (..., n, (n+1)) preceded by (n-1) zeros, as an infinite lower triangular matrix.
5

%I #22 Aug 07 2022 22:13:39

%S 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,

%T 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,

%U 0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0

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

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

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

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

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

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

%H Micah Manary, <a href="/A171608/b171608.txt">Table of n, a(n) for n = 1..5050</a>

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

%e First few rows of the triangle:

%e 1;

%e 2, 0;

%e 0, 2, 0;

%e 0, 3, 0, 0;

%e 0, 0, 3, 0, 0;

%e 0, 0, 4, 0, 0, 0;

%e 0, 0, 0, 4, 0, 0, 0;

%e 0, 0, 0, 5, 0, 0, 0, 0;

%e 0, 0, 0, 0, 5, 0, 0, 0, 0;

%e 0, 0, 0, 0, 6, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0;

%e ...

%p A171609 := proc(n,k)

%p if k = ceil(n/2) then

%p floor( (n+2)/2) ;

%p else

%p 0;

%p end if;

%p end proc:

%p seq(seq( A171609(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Sep 23 2021

%Y Cf. A002620, A084265, A028724, A171608.

%K nonn,tabl,easy

%O 1,2

%A _Gary W. Adamson_, Dec 12 2009

%E More terms from _Micah Manary_, Aug 07 2022