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A101330
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Array read by antidiagonals: T(n,k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 1, k >= 1.
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11
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3, 5, 5, 8, 8, 8, 11, 13, 13, 11, 13, 18, 21, 18, 13, 16, 21, 29, 29, 21, 16, 18, 26, 34, 40, 34, 26, 18, 21, 29, 42, 47, 47, 42, 29, 21, 24, 34, 47, 58, 55, 58, 47, 34, 24, 26, 39, 55, 65, 68, 68, 65, 55, 39, 26, 29, 42, 63, 76, 76, 84, 76, 76, 63, 42, 29, 32, 47
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OFFSET
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1,1
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COMMENTS
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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)).
The Zeckendorf expansion can be written n = Sum_{i=1..k} F(a_i), where a_{i+1} >= a_i + 2. In this formulation, the product becomes: if n = Sum_{i=1..k} F(a_i) and m = Sum_{j=1..l} F(b_j) then n o m = Sum_{i=1..k} Sum_{j=1..l} F(a_i + b_j).
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.
The papers by Arnoux, Grabner et al. and Messaoudi discuss this sequence and generalizations.
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LINKS
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FORMULA
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x o y = 3 x y - x [(y+1)/phi^2] - y [(x+1)/phi^2]. For proof see link. - Fred Lunnon, May 19 2008
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EXAMPLE
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Array begins:
3 5 8 11 13 16 18 21 24 ...
5 8 13 18 21 26 29 34 39 ...
8 13 21 29 34 42 47 55 63 ...
11 18 29 40 47 58 65 76 87 ...
13 21 34 47 55 68 76 89 102 ...
16 26 42 58 68 84 94 110 126 ...
18 29 47 65 76 94 105 123 141 ...
21 34 55 76 89 110 123 144 165 ...
24 39 63 87 102 126 141 165 189 ...
...........................................
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MATHEMATICA
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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]}]]; (* Robert G. Wilson v, Feb 09 2005 *)
Flatten[ Table[ kfp[i, n - i], {n, 2, 13}, {i, n - 1, 1, -1}]] (* Robert G. Wilson v, Feb 09 2005 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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