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
Examples
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 ... ...
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.
Crossrefs
Programs
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Mathematica
Table[#*Binomial[m, 2] + m*Binomial[#, 2] &[n - m + 1], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, May 22 2025 *) -
PARI
T(n,m) = n*binomial(m,2) + m*binomial(n,2)
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).