What OEIS refers to as the "Chow-Long" sequence A078588 --- following the long-established tradition of naming mathematical concepts after people who didn't discover them and quite possibly never even mentioned them --- arises in a problem apparently originating with D. L. Silverman, J. Recr. Math. 9 (4) 208, problem 567 (1976-77). Define a binary sequence of (initially) positive integer n by CL(n) == floor(2*phi*n) - 2*floor(phi*n) where phi denotes the golden section number (1 + sqrt5)/2; the first few values for n > 0 are 1010010110 1001011010 1101001011 0100101001 0110100101 1010110100 1011010010 1001011010 01011010 ... Now partition the positive integers into two sets A = A_0, B = A_1 defined by A_k == { n | CL(n) = k }; so A = { 2, 4, 5, 7, 10, 12, 13, 15, 18, 20, ... }, B = { 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 21, ... }; then form the sets of sums of pairs of distinct(?) elements from each set; then take the complement of their union, the set S "avoided" by the partition: S = { 1, 2, 3, 5, 8, 13, 21, 34, 55, ... }. This turns out to be the Fibonacci numbers. See for example T. Chow, A new characterization of the Fibonacci-free partition, Fibonacci Q. 29 (1991), 174-180; online at http://www-math.mit.edu/~tchow/cv.html [but really, it is actually much easier to prove than that!] While investigating Knuth's "circle" product, I encountered CL(n) in a different setting, which suggested generalising it to the m-ary sequence CL_m(n) == floor(m*phi*n) - m*floor(phi*n); for instance, taking m = 3, the first few values of CL_3(n) for n > 0 are 1021020210 2101021021 2102102021 0210102102 1210210102 1021210210 2021021010 2102121021 0202102101 0210202102 1010210212 1021020210 2101021021 2102102 ... [I posted this sequence a while back, but nobody hacked it ...] Feeding this back into the original context, we define m sets as before A_k == { n | CL(n) = k }, for m = 3, A = {2, 5, 7, 10, 13, 15, 18, 23, 26, 28, 31, ... }, B = {1, 4, 9, 12, 14, 17, 20, 22, 25, 30, 33, ... }, C = {3, 6, 8, 11, 16, 19, 21, 24, 27, 29, 32, ... }; the sets of sums of m-plets of each set; and finally the set S avoided by their union, for m = 3, S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 21, 23, 26, 29, 31, 34, 39, 42, 47, 55, 60, 68, 76, 81, 89, 102, 110, 123, 144, 157, 178, 199, 212, 233, 267, 288, 322, 377, 411, 466, 521, 555, 610, 699, 754, 843, 987, ... }. First problem: conjecture the explicit form of this set for m = 3 (easy); second problem: prove it (hmmm...). Fred Lunnon [20/06/08]