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%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
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