It is known that every number is the sum of three triangular numbers (that was when Gauss said Eureka!), and https://oeis.org/A008443 counts the ways. It is believed that every number is the sum of three palindromes (see A261422 or A261132, which have remarkable graphs). Both triangular numbers and palindromes have about O(sqrt(N)) terms below N. My question is, what is the least dense sequence S that works? One might think that one only needs O(N^(1/3)) terms below N to get S+S+S = Z+ Are there specific examples known of sparser sequences that work, Richard? (I didn't find anything in UPINT) Also, is there a simple proof of Gauss's theorem that doesn't work backwards from the 4-squares theorem? Is there a constructive algorithm that takes a number n and produces three triangular numbers that add to n? Same questions for sums of r squares, for r=2,3,4,... Best regards Neil