Consulting http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence , the section about "The Prouhet-Tarry-Escott problem" I see that "Flanelle's theorem" was discovered previously by Eugene Prouhet in 1851. Also, "my" closed form WIlsonian sum for C^n, also was known previously, I infer from http://mathworld.wolfram.com/Thue-MorseConstant.html Indeed this paper Jean-Paul Allouche and Jeffrey Shallit: The ubiquitous Prouhet-Thue-Morse sequence, Sequences and their applications, Proceedings of SETA'98 (Springer 1999) pages 1-16 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104.476&rep=rep1&ty... discusses that, and in section 52, it gives the solution to David Wilson's sqrt(2)/2 problem, which was proved by David Robbins in problem E2692, Amer. Math'l Monthly 86 (1979) 394-395 solving a puzzle posed by D.R.Woods in AMM 85 (1978) 48 and also solved by Allouche in 1987 as explained in this paper with a short proof. On the other hand, the Flanelle(P) generalization might be new, anyhow was not mentioned in these sources, and the more general problem of compiling a table of "Wilsonian sums" -- which we now see ought to be called "Thue-Morse sums" or "Prouhet sums" -- also might be new (by the same criterion).