Ok, here is a crazy way, owing a bit to internet research

define some "pell numbers" where pell(N) =((1+SQRT(2))^A1-(1-SQRT(2))^A1)/(2*SQRT(2))

starting N at 0 these go:

{P} = 0,1,2,5,12,29,70,169,408,985,2378,5741...

now take differences between these, so that D(N) = P(N+2)-P(N+1)

the differences are:

{D} = 1,3,7,17,41,99,239,577,1393,3363,8119,19601...

now calculate N = (D-1)/2

{N} = 1,3,8,20,49,119,288,696,1681,4059,9800....

not calculate the Nth triangular number

T(N) = N(N+1)/2

{T} = 1,6,36,210,1225,7140,41616,242556,1413721,8239770,48024900

items 1,3,7,9 etc of the above set are triangular numbers which are also squares, items 2,4,6,8 etc are triangular number which are twice another triangular number

so take items 2,4,6,8 of the above set

{T'} = 6,210,7140,242556,8239770

now, for each T calculate

W = sqrt(T + 1/4) + 1/2

this gives

{W} = 3,15,85,493,2871,16731

Now, each number in {W} is a possible number of white balls!