A Remark on Parity Sequences
نویسنده
چکیده
If n = 2, then the set j£ is closely related to the Fibonacci sequence; specifically, t GT2 iff the 7 term of the Fibonacci sequence is odd. We ask, for each w, which numbers are uniquely expressible as the sum of two distinct elements of Tn. In general, for any given w, one can determine exactly which numbers are uniquely expressible. If w = 2, it is easy to see that there are five such numbers: 3 = 1 + 2, 5 = 1 + 4, 7 = 2 + 5, 8 = 1 + 7, and 10 = 2 + 8. If w = 3, then there are exactly eight uniquely expressible numbers: 3 = 1 + 2, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 5, 7 = 2 + 5, 8 = 3 + 5, 9 = 1 + 8, and 16 = 1 + 15. If w = 4, then there are exactly five uniquely expressible numbers: 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, and 16 = 4 + 12. If w>3, then 1,2,3 e Tn, so that 3 and 4 are uniquely expressible. The principal theorem of this note answers this question for all other situations. Let U„ be the set of all integers which are uniquely expressible as the sum of two distinct elements of Tn. Thus, we have just observed that U2 = {3,5,7,8,10}, U3 = {3,4,5,6,7,8,9,16}, and U4 = {3,4,6,8,16}. The following principal theorem characterizes Un for n > 5. Theorem: Let n > 5. Then U„ = {3,4, n n + 3, In In + 4} if n = 2 +1 for some 1, and Un = {3,4} otherwise.
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