A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)*k^2 + n*(k-1) + 1 (k >= 1, n >= 0).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 4, 1, 1, 4, 14, 16, 7, 1, 1, 5, 23, 34, 29, 11, 1, 1, 6, 34, 58, 63, 46, 16, 1, 1, 7, 47, 88, 109, 101, 67, 22, 1, 1, 8, 62, 124, 167, 176, 148, 92, 29, 1, 1, 9, 79, 166, 237, 271, 259, 204, 121, 37, 1, 1, 10, 98, 214, 319, 386, 400, 358, 269, 154, 46, 1, 1, 11, 119, 268, 413, 521, 571, 554, 473, 343, 191, 56, 1

Offset: 0

Author

N. J. A. Sloane, Aug 11 2025

Keywords

• nonn
• tabl

Comments

T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-armed long-legged V's.

Examples

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
   1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 1, 2, 4, 7, 11, 16, 22, 29, ...
   1, 2, 7, 16, 29, 46, 67, 92, 121, ...
   1, 3, 14, 34, 63, 101, 148, 204, 269, ...
   1, 4, 23, 58, 109, 176, 259, 358, 473, ...
   1, 5, 34, 88, 167, 271, 400, 554, 733, ...
   1, 6, 47, 124, 237, 386, 571, 792, 1049, ...
   1, 7, 62, 166, 319, 521, 772, 1072, 1421, ...
    ...
The first few antidiagonals are:
    1,
    1, 1,
    1, 1, 1,
    1, 2, 2, 1,
    1, 3, 7, 4, 1,
    1, 4, 14, 16, 7, 1,
    1, 5, 23, 34, 29, 11, 1,
    1, 6, 34, 58, 63, 46, 16, 1,
    1, 7, 47, 88, 109, 101, 67, 22, 1,
     ...
		

References

• David O. H. Cutler and N. J. A. Sloane, paper in preparation, August 1 2025.

Links

• N. J. A. Sloane, Illustration for T(3,2) = 14.
• N. J. A. Sloane, Illustration for T(3,3) = 34 (a 3-armed V is also known as a Wu).

Crossrefs

This is a companion to the array A386478.

The rows and columns include A000124, A130883, A140064, A383464, and A008865.

Extensions

Under construction, please do not touch.