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A212954 Array of Ramsey numbers R(n,k) (n >= 1, k >= 1) read by antidiagonals. 4

%I #47 Mar 21 2024 16:26:03

%S 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,

%T 25,18,7,1,1,8,23

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

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

%D See A059442.

%H Stanislaw Radziszowski, <a href="https://doi.org/10.37236/21">Small Ramsey Numbers</a>, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1, Mar 3 2017.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamseyNumber.html">Ramsey Number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramsey_theorem">Ramsey's theorem</a>

%F R(r, 1) = R(1, r) = 1

%F R(r, 2) = R(2, r) = r

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

%F 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

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

%e The initial antidiagonals are:

%e 1,

%e 1, 1,

%e 1, 2, 1,

%e 1, 3, 3, 1,

%e 1, 4, 6, 4, 1,

%e 1, 5, 9, 9, 5, 1,

%e 1, 6, 14, 18, 14, 6, 1,

%e 1, 7, 18, 25, 25, 18, 7, 1,

%e 1, 8, 23, ?, ?, ?, 23, 8, 1,

%e 1, 9, 28, ?, ?, ?, ?, 28, 9, 1,

%e 1, 10, 36, ?, ?, ?, ?, ?, 36, 10, 1,

%e ...

%e ...

%Y Cf. A000791, A213368 (row sums).

%K nonn,tabl,hard,more,changed

%O 1,5

%A _Joerg Arndt_, Jun 01 2012

%E Edited by _N. J. A. Sloane_, Nov 05 2023

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)