A393442 Array read by downward antidiagonals: T(n,k) (n>=0, k>=0) is the maximum number of regions the plane can be divided into by drawing n circles and k lines.

1, 2, 2, 4, 4, 4, 7, 8, 8, 8, 11, 13, 14, 14, 14, 16, 19, 21, 22, 22, 22, 22, 26, 29, 31, 32, 32, 32, 29, 34, 38, 41, 43, 44, 44, 44, 37, 43, 48, 52, 55, 57, 58, 58, 58, 46, 53, 59, 64, 68, 71, 73, 74, 74, 74, 56, 64, 71, 77, 82, 86, 89, 91, 92, 92, 92, 67, 76, 84, 91, 97, 102, 106, 109, 111, 112, 112, 112, 79, 89, 98, 106, 113, 119, 124, 128, 131, 133, 134, 134, 134

Offset

0,2

Comments

The individual circles can have any radii; the lines are infinite.

References

Jacob Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349-364. See Eq. (23). [Note that the title uses the old spelling of Teilung. This is not a typo.] Note also that the formula in (23) is wrong if there are no lines, in that case (11) must be used.

Links

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

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.

Formula

T(n,k) = 1+k+2*n*k+k*(k-1)/2+n*(n-1), except T(n,0) = n^2-n+2 if n >= 1.

Example

The array begins:
  1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, ...
  2, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, ...
  4, 8, 14, 21, 29, 38, 48, 59, 71, 84, 98, ...
  8, 14, 22, 31, 41, 52, 64, 77, 91, 106, 122, ...
  14, 22, 32, 43, 55, 68, 82, 97, 113, 130, 148, ...
  22, 32, 44, 57, 71, 86, 102, 119, 137, 156, 176, ...
  32, 44, 58, 73, 89, 106, 124, 143, 163, 184, 206, ...
  44, 58, 74, 91, 109, 128, 148, 169, 191, 214, 238, ...
  58, 74, 92, 111, 131, 152, 174, 197, 221, 246, 272, ...
  74, 92, 112, 133, 155, 178, 202, 227, 253, 280, 308, ...
  92, 112, 134, 157, 181, 206, 232, 259, 287, 316, 346, ...
  ...
The first few downward antidiagonals are:
  1,
  2, 2,
  4, 4, 4,
  7, 8, 8, 8,
  11, 13, 14, 14, 14,
  16, 19, 21, 22, 22, 22,
  22, 26, 29, 31, 32, 32, 32,
  29, 34, 38, 41, 43, 44, 44, 44,
  37, 43, 48, 52, 55, 57, 58, 58, 58,
  46, 53, 59, 64, 68, 71, 73, 74, 74, 74,...

Mathematica

A393442[n_, k_] := n*(n-1) + If[k == 0 && n >= 1, 2, 1 + k + 2*n*k + k*(k-1)/2];
Table[A393442[k, n - k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Mar 30 2026 *)

Crossrefs

Row 0 is A000124, row 1 is essentially A034856; column 0 is both A014206 and A386480, columns 1 and 2 are (apart from their initial values) the same as column 0, and column 3 is essentially A002061.

Cf. A140063 (main diagonal).

Sequence in context: A372678 A120456 A367039 * A115383 A219156 A210036

Adjacent sequences: A393439 A393440 A393441 * A393443 A393444 A393445

Keyword

nonn,tabl

Author

Jonas Karlsson and N. J. A. Sloane, Mar 30 2026

Status

approved