A384125 Array read by antidiagonals: T(n,m) is the number of edges in the n X m rook graph K_n X K_m.

0, 1, 1, 3, 4, 3, 6, 9, 9, 6, 10, 16, 18, 16, 10, 15, 25, 30, 30, 25, 15, 21, 36, 45, 48, 45, 36, 21, 28, 49, 63, 70, 70, 63, 49, 28, 36, 64, 84, 96, 100, 96, 84, 64, 36, 45, 81, 108, 126, 135, 135, 126, 108, 81, 45, 55, 100, 135, 160, 175, 180, 175, 160, 135, 100, 55

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Eric Weisstein's World of Mathematics, Edge Count.

Eric Weisstein's World of Mathematics, Rook Graph.

Formula

T(n,m) = n*binomial(m,2) + m*binomial(n,2).

T(n,m) = binomial(n*m,2) - 2*binomial(n,2)*binomial(m,2).

T(n,m) = T(m,n).

Example

Array begins:
=======================================
n\m |  1  2   3   4   5   6   7   8 ...
----+----------------------------------
  1 |  0  1   3   6  10  15  21  28 ...
  2 |  1  4   9  16  25  36  49  64 ...
  3 |  3  9  18  30  45  63  84 108 ...
  4 |  6 16  30  48  70  96 126 160 ...
  5 | 10 25  45  70 100 135 175 220 ...
  6 | 15 36  63  96 135 180 231 288 ...
  7 | 21 49  84 126 175 231 294 364 ...
  8 | 28 64 108 160 220 288 364 448 ...
  ...

Mathematica

Table[#*Binomial[m, 2] + m*Binomial[#, 2] &[n - m + 1], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, May 22 2025 *)

Prog

(PARI) T(n, m) = n*binomial(m, 2) + m*binomial(n, 2)

Crossrefs

Main diagonal is A045991.

Columns 1..6 are A000217(n-1), A000290, A045943, A054000, A269457(n-1), A067707.

Cf. A003991 (number of vertices), A360855 (triangles), A384120 (all cliques).

Sequence in context: A061800 A394651 A382715 * A218789 A324335 A238162

Adjacent sequences: A384122 A384123 A384124 * A384126 A384127 A384128

Keyword

nonn,tabl,easy

Author

Andrew Howroyd, May 20 2025

Status

approved