

A332963


Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n1,k1)*T(n1,k)+1)/T(n2,k1) for n > 2, 0 < k <= n.


1



1, 2, 2, 1, 5, 1, 2, 3, 3, 2, 1, 7, 2, 7, 1, 2, 4, 5, 5, 4, 2, 1, 9, 3, 13, 3, 9, 1, 2, 5, 7, 8, 8, 7, 5, 2, 1, 11, 4, 19, 5, 19, 4, 11, 1, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 1, 13, 5, 25, 7, 29, 7, 25, 5, 13, 1, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2
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OFFSET

0,2


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FORMULA

By rows: a(2n,0)=a(2n,2n)=1, a(2n+1,0)=a(2n+1,2n+1)=2 for all n >= 0, while the interior numbers are defined recursively by a(n,k) = (a(n1,k1)*a(n1,k)+1)/a(n2,k1) for n >= 2, 0 < k <= n.
By antidiagonals: T(0,2n)=T(2n,0)=1, T(0,2n+1)=T(2n+1,0)=2 for all n >= 0, while the interior numbers are defined recursively by T(r,k) = (T(r1,k)*(Tr,k1)+1)/T(r1,k1) for r,k > 0.


EXAMPLE

For row 3: a(3,0)=2, a(3,1)= 3, a(3,2)=3, a(3,3)=2.
For antidiagonal 3: T(3,0)=2, T(3,1)=7, T(3,2)=5, T(3,3)=13, ...
Triangle begins:
1;
2, 2;
1, 5, 1;
2, 3, 3, 2;
1, 7, 2, 7, 1;
2, 4, 5, 5, 4, 2;
...


PROG

(PARI) T(n, k) = if ((n<0)  (n<k), 0, if ((k==0)  (k==n), if (n%2, 2, 1), (T(n1, k1)*T(n1, k)+1)/T(n2, k1)));
matrix(7, 7, n, k, T(n1, k1)) \\ to see the triangle \\ Michel Marcus, Mar 16 2020


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AUTHOR



STATUS

approved



