A212954 Array of Ramsey numbers R(n,k) (n >= 1, k >= 1) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 14, 18, 14, 6, 1, 1, 7, 18, 25, 25, 18, 7, 1, 1, 8, 23

Offset

1,5

Comments

Essentially the same as A059442, which is the main entry for these numbers.

References

See A059442.

Links

Table of n, a(n) for n=1..39.

Stanislaw Radziszowski, Small Ramsey Numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1, Mar 3 2017.

Eric Weisstein's World of Mathematics, Ramsey Number

Wikipedia, Ramsey's theorem

Formula

R(r, 1) = R(1, r) = 1

R(r, 2) = R(2, r) = r

R(r, s) <= R(r-1, s) + R(r, s-1)

R(r, s) <= R(r-1, s) + R(r, s-1) - 1 if R(r-1, s) and R(r, s-1) are both even

R(r, r) <= 4 * R(r, r-2) + 2

Example

The initial antidiagonals are:
1,
1,  1,
1,  2,  1,
1,  3,  3,  1,
1,  4,  6,  4,  1,
1,  5,  9,  9,  5,  1,
1,  6, 14, 18, 14,  6,  1,
1,  7, 18, 25, 25, 18,  7,  1,
1,  8, 23,  ?,  ?,  ?, 23,  8,  1,
1,  9, 28,  ?,  ?,  ?,  ?, 28,  9,  1,
1, 10, 36,  ?,  ?,  ?,  ?,  ?, 36, 10,  1,
...
...

Crossrefs

Cf. A000791, A213368 (row sums).

Sequence in context: A095142 A180171 A140822 * A299807 A089239 A223968

Adjacent sequences: A212951 A212952 A212953 * A212955 A212956 A212957

Keyword

nonn,tabl,hard,more

Author

Joerg Arndt, Jun 01 2012

Extensions

Edited by N. J. A. Sloane, Nov 05 2023

Status

approved