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A228012 The 2-color Rado number for the equation x_1 + x_2 + ... + x_n = 2*x_0 0
1, 4, 5, 8, 9, 14, 16, 23, 25, 33, 36, 46, 49, 60, 64, 77, 81, 95, 100, 116, 121, 138, 144, 163, 169, 189, 196, 218, 225, 248, 256, 281, 289, 315, 324, 352, 361, 390, 400, 431, 441, 473, 484, 518, 529, 564, 576, 613, 625, 663, 676, 716, 729, 770, 784, 827, 841 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

For n=1, the Rado number is infinity (since the positive integers can be colored using two colors in such a way that no monochromatic solution to the equation x_1 = 2*x_0 exists).

REFERENCES

D. Schaal and D. Vestal, Rado numbers for x_1 + x_2 + ... + x_(m-1) = 2*x_m, Congressus Numerantium, 191(2008), 105-116.

LINKS

Table of n, a(n) for n=2..58.

FORMULA

For n >= 5, a(n) = ceiling(ceiling(n/2)*n/2).

Conjecture: For n >= 5, a(n) = (1-(-1)^n+i*(-i)^n-i*i^n+n-(-1)^n*n+2*n^2)/8, where i=sqrt(-1). G.f.: x^2*(x^9-2*x^7-x^6+x^5+x^4-3*x-1) / ((x-1)^3*(x+1)^2*(x^2+1)). - Colin Barker, Aug 12 2013

EXAMPLE

For n=4, we have a(4) = 5, meaning that the 2-color Rado number for the equation E:x_1 + x_2 + x_3 + x_4 = 2*x_0 is 5.  The coloring (or partition) Red = {1,4} and Blue = {2,3} avoids a monochromatic solution to E; however, any 2-coloring of the integers {1,2,3,4,5} will have a monochromatic solution to E.

CROSSREFS

Sequence in context: A020668 A020934 A094004 * A067271 A268128 A064394

Adjacent sequences:  A228009 A228010 A228011 * A228013 A228014 A228015

KEYWORD

easy,nonn

AUTHOR

Donald Vestal, Aug 08 2013

STATUS

approved

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)