A135090 -id:A135090 - cp's OEIS frontend

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.

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

Offset: 1

N. J. A. Sloane, Jan 25 2005

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.

			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 ...
  ...........................................

T. D. Noe, First 100 antidiagonals, flattened
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, Propriétés arithmétiques et dynamiques du fractal de Rauzy [alternative copy]
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.

See A101646 and A135090 for other versions.
Main diagonal is A101332.
Rows: A026274 (row 1), A101345 (row 2), A101642 (row 3).
Cf. A035517, A014417, A060144.
Cf. A101385, A101633, A101858 for related definitions of product.

h := n -> floor(2*(n + 1)/(sqrt(5) + 3)):  # A060144(n+1)
T := (n, k) -> 3*n*k - n*h(k) - k*h(n):
seq(print(seq(T(n, k), k = 1..9)), n = 1..7);  # Peter Luschny, Mar 21 2024

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 *)
A101330[n_, k_]:=3*n*k-n*Floor[(k+1)/GoldenRatio^2]-k*Floor[(n+1)/GoldenRatio^2];
Table[A101330[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

T(n, k) = 3*n*k - n*floor((k+1)/phi^2) - k*floor((n+1)/phi^2). For proof see link. - Fred Lunnon, May 19 2008

T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = A060144(n + 1). - Peter Luschny, Mar 21 2024

More terms from David Applegate, Jan 26 2005

A356592 Array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = Sum_{i, j >= 3} t_i * u_j * T(i+j) where Sum_{i >= 3} t_i * T(i) and Sum_{j >= 3} u_j * T(j) are the greedy tribonacci representations of n and k, respectively, and T = A000073.

Original entry on oeis.org

0, 0, 0, 0, 7, 0, 0, 13, 13, 0, 0, 20, 24, 20, 0, 0, 24, 37, 37, 24, 0, 0, 31, 44, 57, 44, 31, 0, 0, 37, 57, 68, 68, 57, 37, 0, 0, 44, 68, 88, 81, 88, 68, 44, 0, 0, 51, 81, 105, 105, 105, 105, 81, 51, 0, 0, 57, 94, 125, 125, 136, 125, 125, 94, 57, 0

Offset: 0

Rémy Sigrist, Sep 11 2022

This sequence is to tribonacci numbers (A000073) what A135090 is to Fibonacci numbers (A000045).

			Array A(n, k) begins:
  n\k | 0   1    2    3    4    5    6    7    8    9   10
  ----+---------------------------------------------------
    0 | 0   0    0    0    0    0    0    0    0    0    0
    1 | 0   7   13   20   24   31   37   44   51   57   64
    2 | 0  13   24   37   44   57   68   81   94  105  118
    3 | 0  20   37   57   68   88  105  125  145  162  182
    4 | 0  24   44   68   81  105  125  149  173  193  217
    5 | 0  31   57   88  105  136  162  193  224  250  281
    6 | 0  37   68  105  125  162  193  230  267  298  335
    7 | 0  44   81  125  149  193  230  274  318  355  399
    8 | 0  51   94  145  173  224  267  318  369  412  463
    9 | 0  57  105  162  193  250  298  355  412  460  517
   10 | 0  64  118  182  217  281  335  399  463  517  581

A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A000073, A101330, A135090.

PARI
```
See Links section.
```

A(n, 0) = A(0, k) = 0.
A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k) for m, n, k >= 5.

A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 2, 1, 0, 3, 3, 3, 3, 2, 0, 5, 5, 5, 4, 3, 2, 0, 8, 8, 8, 7, 5, 4, 3, 0, 13, 13, 13, 11, 8, 6, 4, 3, 0, 21, 21, 21, 18, 13, 10, 7, 5, 3, 0, 34, 34, 34, 29, 21, 16, 11, 8, 6, 4, 0, 55, 55, 55, 47, 34, 26, 18, 13, 9, 6, 4

Offset: 0

Peter Munn, Sep 23 2022

Note the Zeckendorf representation of 0 is taken to be the empty sum.
The Wythoff array A035513 is the subtable formed by rows 3, 11, 16, 24, 32, ... (A035337). If, instead, we use rows 2, 7, 10, 15, 20, ... (A035336) or 1, 4, 6, 9, 12, ... (A003622), we get the Wythoff array extended by 1 column (A287869) or 2 columns (A287870) respectively.
Similarly, using A035338 truncates by 1 column; and in general if S_k is column k of the Wythoff array then the rows here numbered by S_k form an array A_k that starts with column k-2 of the Wythoff array. (A_0 and A_1 are the 2 extended arrays mentioned above.) As every positive integer occurs exactly once in the Wythoff array, every row except row 0 of A(.,.) is a row of exactly one such A_k.
Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).
For k > 0, the first row to contain k is A348853(k).

			Example for n = 4, k = 3. The Zeckendorf representation of 4 is F_4 + F_2 = 3 + 1. So the values of i in the sums in the definition are 4 and 2; hence A(4,3) = Sum_{i = 2,4} F_{i+k-2} = F_{4+3-2} + F_{2+3-2} = F_5 + F_3 = 5 + 2 = 7.
Square array A(n,k) begins:
   n\k| 0   1    2    3    4    5    6
  ----+--------------------------------
   0  | 0   0    0    0    0    0    0  ...
   1* | 0   1    1    2    3    5    8  ...
   2  | 1   1    2    3    5    8   13  ...
   3  | 1   2    3    5    8   13   21  ...
   4* | 1   3    4    7   11   18   29  ...
   5  | 2   3    5    8   13   21   34  ...
   6* | 2   4    6   10   16   26   42  ...
   7  | 3   4    7   11   18   29   47  ...
   8  | 3   5    8   13   21   34   55  ...
   9* | 3   6    9   15   24   39   63  ...
  10  | 4   6   10   16   26   42   68  ...
  11  | 4   7   11   18   29   47   76  ...
  12* | 4   8   12   20   32   52   84  ...
  ...
The asterisked rows form the start of the extended Wythoff array (A287870).

Encyclopedia of Mathematics, Zeckendorf representation

Columns, some differing initially: A005206 (1), A022342 (3), A026274 (4), A101345 (5), A101642 (6).
Rows: A000045 (1), A000204 (4).
Related to subtable A287870 as A130128 (as a square) is to A054582.
Other subtables: A035513, A287869.
See the comments for the relationship to A003622, A035336, A035337, A035338, A348853.
See the formula section for the relationship to A003714, A022342, A135090, A356874.

A5206(m) = if(m>0,m-A5206(A5206(m-1)),0)
A(n,k) = if(k==2,n, if(k==1,A5206(n), if(k==0,n-A5206(n), A(n,k-2)+A(n,k-1)))) \\ simple encoding of formulas, not efficient

For n >= 0, k >= 0 unless stated otherwise:
A(n,k) = A356874(floor(A003714(n)*2^(k-1))).
A(n,1) = A005206(n).
A(n,2) = n.
A(n,k+2) = A(n,k) + A(n,k+1).
A(A022342(n+1),k) = A(n,k+1).
For k >= 4, A(n,k) = A135090(n,A000045(k-2)).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A356592 Array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = Sum_{i, j >= 3} t_i * u_j * T(i+j) where Sum_{i >= 3} t_i * T(i) and Sum_{j >= 3} u_j * T(j) are the greedy tribonacci representations of n and k, respectively, and T = A000073.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula