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A101330 Array read by antidiagonals: T(n,k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 1, k >= 1. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]
Vincent Canterini and Anne Siegel, Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144.
P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90.
D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
W. F. Lunnon, Proof of formula
A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy, Journal de théorie des nombres de Bordeaux, 10 no. 1 (1998), p. 135-162.
A. Messaoudi, Généralisation de la multiplication de Fibonacci, Math. Slovaca, 50 (2) (2000), 135-148.
A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.
FORMULA
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
EXAMPLE
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 ...
...........................................
MATHEMATICA
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 *)
CROSSREFS
See A101646 and A135090 for other versions.
Cf. A035517, A014417. See A101385, A101633, A101858 for related definitions of product.
Main diagonal is A101332. First row equals A026274. Second row is A101345. Third row is A101642.
Sequence in context: A021285 A138575 A263209 * A063285 A316938 A112507
KEYWORD
nonn,tabl,easy,nice
AUTHOR
N. J. A. Sloane, Jan 25 2005
EXTENSIONS
More terms from David Applegate, Jan 26 2005
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)