A393443 Array read by downward antidiagonals: T(n,k) (n>=0, k>=0) = 1+k+2*n*k+k*(k-1)/2+n*(n-1).

1, 2, 1, 4, 4, 3, 7, 8, 8, 7, 11, 13, 14, 14, 13, 16, 19, 21, 22, 22, 21, 22, 26, 29, 31, 32, 32, 31, 29, 34, 38, 41, 43, 44, 44, 43, 37, 43, 48, 52, 55, 57, 58, 58, 57, 46, 53, 59, 64, 68, 71, 73, 74, 74, 73, 56, 64, 71, 77, 82, 86, 89, 91, 92, 92, 91, 67, 76, 84, 91, 97, 102, 106, 109, 111, 112, 112, 111, 79, 89, 98, 106, 113, 119, 124, 128, 131, 133, 134, 134, 133

Offset

0,2

Comments

Note that this is not quite the same as A393442, which is the maximum number of regions the plane can be divided into by drawing n circles and k lines (the two arrays differ in the k=0 column).

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.]

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

G.f.: (1 - y + y^2 - x*(2 - 4*y + 4*y^2) + x^2*(3 - 7*y + 5*y^2))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Mar 30 2026

Example

The array begins:
  1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, ...
  1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, ...
  3, 8, 14, 21, 29, 38, 48, 59, 71, 84, 98, ...
  7, 14, 22, 31, 41, 52, 64, 77, 91, 106, 122, ...
  13, 22, 32, 43, 55, 68, 82, 97, 113, 130, 148, ...
  21, 32, 44, 57, 71, 86, 102, 119, 137, 156, 176, ...
  31, 44, 58, 73, 89, 106, 124, 143, 163, 184, 206, ...
  43, 58, 74, 91, 109, 128, 148, 169, 191, 214, 238, ...
  57, 74, 92, 111, 131, 152, 174, 197, 221, 246, 272, ...
  73, 92, 112, 133, 155, 178, 202, 227, 253, 280, 308, ...
  91, 112, 134, 157, 181, 206, 232, 259, 287, 316, 346, ...
  ...
The first few downward antidiagonals are:
  1,
  2, 1,
  4, 4, 3,
  7, 8, 8, 7,
  11, 13, 14, 14, 13,
  16, 19, 21, 22, 22, 21,
  22, 26, 29, 31, 32, 32, 31,
  29, 34, 38, 41, 43, 44, 44, 43,
  37, 43, 48, 52, 55, 57, 58, 58, 57,
  46, 53, 59, 64, 68, 71, 73, 74, 74, 73,
  ...

Mathematica

A393443[n_, k_] := 1 + k + 2*n*k + k*(k-1)/2 + n*(n-1) ;
Table[A393443[k, n - k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Mar 31 2026 *)

Crossrefs

Cf. A393442.

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

Cf. A140063 (main diagonal).

Sequence in context: A091335 A362865 A274883 * A140946 A008741 A381299

Adjacent sequences: A393440 A393441 A393442 * A393444 A393445 A393446

Keyword

nonn,tabl

Author

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

Extensions

Data corrected by Paolo Xausa, Mar 31 2026

Status

approved