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A279032 a(n) is the greatest integer such that binomial(a(n),n)*2^(1 - binomial(n,2)) < 1.

%I A279032 #18 Dec 10 2016 03:28:20
%S A279032 0,1,3,6,11,17,27,42,65,100,152,231,349,527,792,1186,1771,2639,3923,
%T A279032 5817,8609,12715,18747,27595,40557,59522,87239,127704,186721,272717,
%U A279032 397913,580029,844734,1229199,1787215,2596587,3769796,5469375,7930078,11490820,16640682
%N A279032 a(n) is the greatest integer such that binomial(a(n),n)*2^(1 - binomial(n,2)) < 1.
%C A279032 a(n) is a lower bound on the Ramsey number R(n,n). In other words, a(n) is less than the least integer N such that every graph on N vertices contains an n clique or a size n independent set.
%D A279032 D. B. West, Introduction to Graph Theory, Pearson, 2015, page 385.
%H A279032 Chai Wah Wu, <a href="/A279032/b279032.txt">Table of n, a(n) for n = 1..1000</a>
%F A279032 a(n) is asymptotic to n*2^(n/2)/(e*sqrt(2)).
%p A279032 f:= proc(n) local k,r,B;
%p A279032   k:= max(floor(n*2^(n/2)/(exp(1)*sqrt(2))),n);
%p A279032   r:= 2^(binomial(n,2)-1);
%p A279032   B:= binomial(k,n);
%p A279032   if B < r then
%p A279032     while B*(k+1)/(k+1-n) < r do k:= k+1; B:= B*k/(k-n) od;
%p A279032   else
%p A279032     while B*(k-1)/k > r do B:= B*(k-1)/k; k:= k-1 od;
%p A279032     k:= k-1;
%p A279032   fi;
%p A279032   k
%p A279032 end proc:
%p A279032 map(f, [$1..40]); # _Robert Israel_, Dec 07 2016
%t A279032 Table[Position[Table[Binomial[m, n] < 2^(Binomial[n, 2] - 1), {m, 1, 50000}],False][[1]] - 1, {n, 1, 25}] // Flatten
%Y A279032 Cf. A212954.
%K A279032 nonn
%O A279032 1,3
%A A279032 _Geoffrey Critzer_, Dec 03 2016
%E A279032 More terms from _Robert Israel_, Dec 07 2016