Column[Table[IntegerPartitions[n,{k}]//Length,{n,1,20},{k,1,n}]]

T[n_,k_]:=T[n,k]=If[n>0&&k>0,T[n-1,k-1]+T[n-k,k],Boole[n==0&&k==0]]
p=Table[T[n,k],{n,1,40},{k,1,40}];
MatrixForm@p

Table[(2m-1)*p[[2*m-6+n,m]],{m,3,15},{n,m,15}]

Fence Conjecture: Ramsey Number R(m,n)=(2m-1)*p(2m-6+n,m)+{1,m,m+1}, for 3<=m<=n.
Here p(n,k) denote both the number of partitions of n into exactly k parts (that is, sums of k positive integers that add to n), and the number of partitions of n into parts of maximum size exactly k. see https://en.wikipedia.org/wiki/Triangle_of_partition_numbers .

We know R(m,n)<=R(m-1,n)+R(m,n-1). The below code verify this inequality for R(3,n). we have R(2,n)=n and n>=5*(p(n,3)-p(n-1,3)) - 2

Table[{n,5*(Length@IntegerPartitions[n,{3}]-Length@IntegerPartitions[n-1,{3}])},{n,150}]

If the conjecture is true, then it must have 5P(n,3)+{1,3,4}=R(3,n) >=  R(4,n)-R(4,n-1) = 7(P(n+2,4)-P(n+1,4)）+{1,4,5}-{1,4,5}. And this inequality holds indeed:

P[n_,k_]:=Length@IntegerPartitions[n,{k}]
Table[{5P[n,3],7(P[n+2,4]-P[n+1,4])},{n,30}]