WFL said: 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 Me: Isn't that Knuth's product? Remember that arrays are stored in the OEIS by antidiabgonals. Neil %I A101330 %S A101330 3,5,5,8,8,8,11,13,13,11,13,18,21,18,13,16,21,29,29,21,16,18,26,34, %T A101330 40,34,26,18,21,29,42,47,47,42,29,21,24,34,47,58,55,58,47,34,24,26, %U A101330 39,55,65,68,68,65,55,39,26,29,42,63,76,76,84,76,76,63,42,29,32,47 %N A101330 Array read by antidiagonals: T(n,k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"). %C A101330 Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). (The eps(i) are 0 or 1, and no two consecutive eps(i) are both 1.) Then the Fibonacci (or circle) product of n and k is n o k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j} (= T(n,k)). %C A101330 The Zeckendorf expansion can be written n=sum_{1<=i<=k} F(a_i), where a_{i+1} >= a_i + 2. In this formulation, the product becomes: if n = sum_{1<=i<=k} F(a_i) and m = sum_{1<=j<=l} F(b_j) then n o m = sum_{i=1}^k sum_{j=1}^l F(a_i + b_j). %C A101330 Knuth shows that this multiplication is associative. This is not true if we change the product to n x k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2}, see A101646. Of course 1 is not a multiplicative identity here, whereas it is in A101646. %C A101330 The papers by Arnoux, Grabner et al. and Messaoudi discuss this sequence and generalizations. %D A101330 P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320. %D A101330 P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90. %D A101330 D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60. %D A101330 A. Messaoudi, Generalisation de la multiplication de Fibonacci, Math. Slovaca, 50 (2) (2000), 135-148. %D A101330 A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985. %H A101330 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b101330.txt">Rows n=1..100 of array, flattened</a> %H A101330 Ali Messaoudi, <a href="http://almira.math.u-bordeaux.fr/jtnb/1998-1/messaoudi.ps">Title?</a> %H A101330 Vincent Canterini and Anne Siegel, <a href="http://www.ams.org/tran/2001-353-12/S0002-9947-01-02797-0/home.html">Geometric representation of substitutions of Pisot type</a>, Trans. Amer. Math. Soc. 353 (2001), 5121-5144. %e A101330 Array begins: %e A101330 3 5 8 11 13 ... %e A101330 5 8 13 18 ... %e A101330 8 13 21 ... %e A101330 11 18 ... %e A101330 13 ... %t A101330 zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[i + j + 2], {i, Length[y]}, {j, Length[z]}]]; (from Robert G. Wilson v Feb 09 2005) %t A101330 Flatten[ Table[ kfp[i, n - i], {n, 2, 13}, {i, n - 1, 1, -1}]] (from Robert G. Wilson v Feb 09 2005) %Y A101330 Cf. A035517, A014417. See A101385, A101633, A101858 for related definitions of product. %Y A101330 Main diagonal is A101332. First row appears to equal A026274 - proof? Second row is A101345. Third row is A101642. %K A101330 nonn,tabl,easy,nice %O A101330 1,1 %A A101330 njas, Jan 25 2005 %E A101330 More terms from David Applegate (david(AT)research.att.com), Jan 26 2005