A228012 The 2-color Rado number for the equation x_1 + x_2 + ... + x_n = 2*x_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

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