Let the base-F (Zeckendorf) representations of integers x, y be x = \sum_i x_i F_i , y = \sum_i y_i F_i ; then the circle and arroba products are resp. x o y = \sum_{i,j} x_i y_j F_{i+j} , x @ y = \sum_{i,j} x_i y_j F_{i+j-2} . x @ y is just x o y shifted towards F_0 by two base-F places. It is no longer associative: for instance at x = 7, y = 4, z = 4, the three possible products take distinct values (x @ y) @ z = 91, x @ (y @ z) = 87, x @ y @ z = 90. To appreciate the way arrosa products interact with circle products, I'm afraid it's probably necessary to wade through some of the (latest) proofs ... Multiplication table for x o y :--- __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 Multiplication table x @ y :--- __y__0__1__2__3__4__5__6__7__8__9__10__11__12 _x _0___0__0__0__0__0__0__0__0__0__0___0___0___0 _1___0__1__2__3__4__5__6__7__8__9__10__11__12 _2___0__2__3__5__7__8_10_11_13_15__16__18__20 _3___0__3__5__8_11_13_16_18_21_24__26__29__32 _4___0__4__7_11_15_18_22_25_29_33__36__40__44 _5___0__5__8_13_18_21_26_29_34_39__42__47__52 _6___0__6_10_16_22_26_32_36_42_48__52__58__64 _7___0__7_11_18_25_29_36_40_47_54__58__65__72 _8___0__8_13_21_29_34_42_47_55_63__68__76__84 _9___0__9_15_24_33_39_48_54_63_72__78__87__96 10___0_10_16_26_36_42_52_58_68_78__84__94_104 11___0_11_18_29_40_47_58_65_76_87__94_105_116 12___0_12_20_32_44_52_64_72_84_96_104_116_128 [Incidentally, negative integers can also be represented in base-F, analogously to binary representation, by equipping them with an infinite tail of odd- or even-indexed F_k . For instance -2 by F_4 + F_6 + F_8 + ... -1 by F_3 + F_5 + F_7 + ... --- begging the mischievous question of what is represented by F_2 + F_4 + F_6 + ... ] WFL On 5/27/08, Dan Asimov <dasimov@earthlink.net> wrote:
Fred wrote:
[various neat formulas for x o y, x @ y, and x o y o z]
Alas, I forget the definition of x @ y.
Fred, could you please remind me?
Thanks,
Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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