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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. 10
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_{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).

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.

REFERENCES

P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.

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.

A. Messaoudi, Generalisation de la multiplication de Fibonacci, Math. Slovaca, 50 (2) (2000), 135-148.

A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.

LINKS

T. D. Noe, Rows n=1..100 of array, flattened

Vincent Canterini and Anne Siegel, Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144.

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, Propriétés arithmétiques et dynamiques du fractal de Rauzy [alternative copy]

FORMULA

x o y = 3 x y - x [(y+1)/phi^2] - y [(x+1)/phi^2]. For proof see link. - W. F. 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]}]]; (from Robert G. Wilson v Feb 09 2005)

Flatten[ Table[ kfp[i, n - i], {n, 2, 13}, {i, n - 1, 1, -1}]] (from 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: A019632 A021285 A138575 * A063285 A112507 A229428

Adjacent sequences:  A101327 A101328 A101329 * A101331 A101332 A101333

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 August 21 15:23 EDT 2014. Contains 245856 sequences.