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 A212954 Triangle read by rows: T(n,k) = R(r,s) (two color Ramsey numbers). 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The triangle is symmetric. Same as A059442 (a square array and omitting the ones). REFERENCES See A059442. LINKS Stanislaw Radziszowski, Small Ramsey Numbers (survey). 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 Triangle begins: 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, ... Also: .                     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. Cf. 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 STATUS approved

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Last modified July 17 08:40 EDT 2019. Contains 325097 sequences. (Running on oeis4.)