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Square array read by antidiagonals: T(n,k) in which if k = j^2 then column k lists j interleaved with j-1 zeros. Otherwise if k = j^2 + j then column k lists the positive integers, starting with j+1, interleaved with j-1 zeros. Otherwise column k lists only zeros. n>=1, j>=1, k>=1.
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%I #20 Feb 04 2013 22:56:41

%S 1,1,2,1,3,0,1,4,0,2,1,5,0,0,0,1,6,0,2,0,3,1,7,0,0,0,0,0,1,8,0,2,0,4,

%T 0,0,1,9,0,0,0,0,0,0,3,1,10,0,2,0,5,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,

%U 12,0,2,0,6,0,0,3,0,0,4,1,13,0,0,0,0,0,0,0,0,0,0,0

%N Square array read by antidiagonals: T(n,k) in which if k = j^2 then column k lists j interleaved with j-1 zeros. Otherwise if k = j^2 + j then column k lists the positive integers, starting with j+1, interleaved with j-1 zeros. Otherwise column k lists only zeros. n>=1, j>=1, k>=1.

%C The number of positive terms in the n-th antidiagonal equals the number of divisors of n.

%C The sum of the elements of n-th antidiagonal equals the sum of divisors of n.

%C All terms of column k are zeros iff k is not a quarter-square A002620.

%C If only the first two elements are positive integers in the n-th antidiagonal then n is prime.

%e First 24 elements of first 8 rows of the square array are

%e 1,2,0,2,0,3,0,0,3,0,0,4,0,0,0,4,0,0,0,5,0,0,0,0,...

%e 1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...

%e 1,4,0,2,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...

%e 1,5,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,...

%e 1,6,0,2,0,5,0,0,0,0,0,0,0,0,0,4,0,0,0,6,0,0,0,0,...

%e 1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...

%e 1,8,0,2,0,6,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,...

%e 1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...

%Y Antidiagonal sums give A000203.

%Y Cf. A000005, A002620, A210959, A221641.

%K nonn,tabl

%O 1,3

%A _Omar E. Pol_, Jan 31 2013