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(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

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

Offset

0,2

Links

Table of n, a(n) for n=0..77.

Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019, Figure 10 p. 4.

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(n-1,k-1)*a(n-1,k)+1)/a(n-2,k-1) 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(r-1,k)*(Tr,k-1)+1)/T(r-1,k-1) 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(n-1, k-1)*T(n-1, k)+1)/T(n-2, k-1)));
matrix(7, 7, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Mar 16 2020

Crossrefs

Cf. A077028, A309555, A309557, A332790, A309559

Sequence in context: A192456 A226948 A010243 * A203953 A326933 A123398

Adjacent sequences: A332960 A332961 A332962 * A332964 A332965 A332966

Keyword

nonn,tabl

Author

Philip K Hotchkiss, Mar 04 2020

Status

approved