A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2*n^2/2 - (3*k-4)*n/2 + 1 (k >= 1, n >= 0).

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 7, 7, 1, 1, 5, 14, 16, 11, 1, 1, 8, 25, 34, 29, 16, 1, 1, 12, 40, 61, 63, 46, 22, 1, 1, 17, 59, 97, 113, 101, 67, 29, 1, 1, 23, 82, 142, 179, 181, 148, 92, 37, 1, 1, 30, 109, 196, 261, 286, 265, 204, 121, 46, 1, 1, 38, 140, 259, 359, 416, 418, 365, 269, 154, 56, 1, 1, 47, 175, 331, 473, 571, 607, 575, 481, 343, 191, 67, 1

Offset: 0

Author

N. J. A. Sloane, Jul 24 2025

Keywords

• nonn
• tabl

Comments

A k-chain is a planar graph consisting of a continuous path made up of k straight segments. T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-chains.

The array is almost symmetric: the difference between T(k,n) and T(n,k) is 2*|k-n| (which is exactly the difference between the numbers of infinite regions). All the rows and columns satisfy the recurrence u(n) = 3*u(n-1) - 3*u(n-2) + u(n-3).

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, 37, ...
  1, 2, 7, 16, 29, 46, 67, 92, 121, ...
  1, 3, 14, 34, 63, 101, 148, 204, 269, ...
  1, 5, 25, 61, 113, 181, 265, 365, 481, ...
  1, 8, 40, 97, 179, 286, 418, 575, 757, ...
  1, 12, 59, 142, 261, 416, 607, 834, 1097, ...
  1, 17, 82, 196, 359, 571, 832, 1142, 1501, ...
  1, 23, 109, 259, 473, 751, 1093, 1499, 1969, ...
  ...
The first few antidiagonals are:
  1,
  1, 1,
  1, 2, 1,
  1, 2, 4, 1,
  1, 3, 7, 7, 1,
  1, 5, 14, 16, 11, 1,
  1, 8, 25, 34, 29, 16, 1,
  1, 12, 40, 61, 63, 46, 22, 1,
  ...
		

References

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

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Crossrefs

The first few rows are A000124, A130883, A140064, A080856, A383465.

The n=1 and 2 columns are A152948 and A386479.

Programs

Mathematica

A386478[k_, n_] := If[k == 0, 1, ((k*n - 3)*k + 4)*n/2 + 1];
Table[A386478[k - n, n], {k, 0, 12}, {n, 0, k}] (* Paolo Xausa, Jul 26 2025 *)

Extensions

Row 0 added by N. J. A. Sloane, Jul 26 2025