A212954 - OEIS

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A212954

Triangle read by rows: T(n,k) = R(r,s) (two color Ramsey numbers).

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	`Table of n, a(n) for n=1..39.` `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 November 13 02:59 EST 2019. Contains 329085 sequences. (Running on oeis4.)