[Allowing x(*)0 = x was a dumb idea, which subsequently turns out to create exceptions to explicit expressions: better to retain the original x(*)0 = 0, by redefining in base-10, 1 = ...0101 in base-F" ; in base-10, 0 = ...0000 in base-F" ; with all other natural numbers ending ...00 . Our third attempt at multiplication x(*)y looks like ____y__0___1___2___3___4____5____6____7____8____9 __x______________________________________________ __0____0___0___0___0___0____0____0____0____0____0 __1____0___3___5___8__11___13___16___18___21___24 __2____0___5___8__13__18___21___26___29___34___39 __3____0___8__13__21__29___34___42___47___55___63 __4____0__11__18__29__40___47___58___65___76___87 __5____0__13__21__34__47___55___68___76___89__102 __6____0__16__26__42__58___68___84___94__110__126 __7____0__18__29__47__65___76___94__105__123__141 __8____0__21__34__55__76___89__110__123__144__165 __9____0__24__39__63__87__102__126__141__165__189 remaining associative without exception.] Looking at column x = 1, it becomes quickly apparent that 1(*)y approximates tau^2*y; with some algebra, it can be shown that explicitly 1(*)y = [tau^2 y + 1/tau] = floor(tau^2*y + 1/tau). [This is just the 2-place left-shift mentioned in connection with Penrose tiling inflation.] Column x = 2 is harder to crack. Subtracting [tau^3 y] gives a sequence with just three values, and tinkering with the offset leads to the binary sequence f(y) = 2(*)y - [tau^3 y - 1/tau^3] = 2(*)y - floor(tau^3*y - 1/tau^3) = 0101101001 0110100101 0010110100 1011010110 1001011010 0101001011 0100101101 0110100101 1010010110 1011010010 ... Sniffing around in OEIS soon turns up A078588, which I'll christen the Chow-Long sequence, with the handy explicit formula CL(y) = 2*{y tau} - {2 y tau} = 2*frac(y*tau)-frac(2*y*tau). Unfortunately, CL(y) is not our sequence f(y). They are locally equal, in the sense that every finite segment of either sequence occurs infinitely often in the other, but globally they are distinct: for instance, the segments (f(0),...,f(70)) = (CL(18),...,CL(89)) coincide maximally. More generally, the earliest occurrence of the initial segment (f(0),...,f(w-1)) occurs at (CL(z),...,CL(z+w-1)), where z = F_{3k}/2 + 1 , w = F_{3k+2} - y . Although the sequences are not integer shifts of one another, their quasi- periodicity and the previous observation permit a striking irrational shift relation to emerge: f(y) = CL(y + 1/(2*tau^4)) ; whence finally 2(*)y = [tau^3 y - 1/tau^3] + CL(y + 1/(2*tau^4)) . Column x = 3 may be reduced in a similar fashion to the ternary sequence g(y) = 3(*)y - [tau^4 y + 1/tau^4] = 1201202012 0121201201 0120120201 2012120120 1012012120 1201012012 0201201212 0120101201 2020120121 2012020120 1212012010 1201202012 0121201201 0120120201 2010120120 2012012120 1201012012 0201201212 0120101201 2120120101 2012020120 1212012010 1201202012 0121 ... which, needless to say, does not appear in OEIS. Well, anyone got any ideas? [Looking in Knuth, perhaps?] Fred Lunnon