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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

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.)