A101330 -id:A101330 - cp's OEIS frontend

A101385 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330).

3, 8, 8, 21, 34, 21, 24, 144, 144, 24, 55, 152, 987, 152, 55, 58, 610, 1008, 1008, 610, 58, 63, 618, 6765, 1032, 6765, 618, 63, 144, 644, 6786, 6820, 6820, 6786, 644, 144, 147, 2584, 6909, 6844, 75025, 6844, 6909, 2584, 147, 152, 2592, 46368, 6972, 75080

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 product of n and k is defined here to be Sum_{i,j} eps(i)*eps(j) Fib_{i*j} (= T(n,k)).

			Array begins:
3 8 21 24 55 ...
8 34 144 152 ...
21 144 987 ...
24 152 ...
55 ...

D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.

Cf. A101330, A035517, A014417. Main diagonal is A101633.

First 3 rows give A101643, A101644, A101645.

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]]; kfpv[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[(i + 1)(j + 1)], {i, Length[y]}, {j, Length[z]}]]; (* Robert G. Wilson v, Feb 09 2005 *)
Flatten[ Table[ kfpv[i, n - i], {n, 2, 12}, {i, n - 1, 1, -1}]] (* Robert G. Wilson v, Feb 09 2005 *)

More terms from David Applegate, Jan 26 2005

A101646 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330). Sometimes called the "arroba" product.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 5, 6, 8, 11, 11, 8, 6, 7, 10, 13, 15, 13, 10, 7, 8, 11, 16, 18, 18, 16, 11, 8, 9, 13, 18, 22, 21, 22, 18, 13, 9, 10, 15, 21, 25, 26, 26, 25, 21, 15, 10, 11, 16, 24, 29, 29, 32, 29, 29, 24, 16, 11, 12, 18, 26, 33, 34, 36, 36, 34, 33, 26, 18, 12

Offset: 1

David Applegate and N. J. A. Sloane, Jan 26 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 product of n and k is defined here to be n x k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2} (= T(n,k)). [Comment corrected by R. J. Mathar, Aug 07 2007]

Although now 1 is the multiplicative identity, in contrast to A101330, this multiplication is not associative. For example, as pointed out by Grabner et al., we have (4 x 7 ) x 9 = 25 x 9 = 198 but 4 x (7 x 9 ) = 4 x 54 = 195.

			Array begins:
  1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...
  2 3  5  7  8 10 11 13 15 16 18 20 21 23 24 26 28 29 31 ...
  3 5  8 11 13 16 18 21 24 26 29 32 34 37 39 42 45 47 50 ...
  4 7 11 15 18 22 25 29 33 36 40 44 47 51 54 58 62 65 69 ...
  5 8 13 18 21 26 29 34 39 42 47 52 55 60 63 68 73 76 81 ...
...

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

Cf. A101330, A101385, A035517, A014417. Main diagonal is A101711.

First 4 rows give A000027, A022342, A026274, A101741.

T[n_, k_] := With[{phi2 = GoldenRatio^2}, n k - Floor[(k + 1)/phi2] Floor[ (n + 1)/phi2]];
Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2020 *)

T(n, k) = my(phi2 = ((1+sqrt(5))/2)^2); n*k - floor((k+1)/phi2)*floor((n+1)/phi2); \\ Michel Marcus, Mar 29 2016

T(n, k) = n*k - [(k+1)/phi^2] [(n+1)/phi^2]. For proof see link. - Fred Lunnon, May 24 2008

A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

Original entry on oeis.org

5, 10, 10, 13, 20, 13, 18, 26, 26, 18, 23, 36, 34, 36, 23, 26, 46, 47, 47, 46, 26, 31, 52, 60, 65, 60, 52, 31, 34, 62, 68, 83, 83, 68, 62, 34, 39, 68, 81, 94, 106, 94, 81, 68, 39, 44, 78, 89, 112, 120, 120, 112, 89, 78, 44, 47, 88, 102, 123, 143, 136, 143, 123, 102, 88, 47, 52

Offset: 1

N. J. A. Sloane, Jan 28 2005

Row 1 / column 1 (given in A101868) = positions of 1 in A188009, viz.,
  A188009 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, ...), A101868 = (5, 10, 13, 18, 23, 26, 31, 34, 39, 44, 47, 52, 57, ...). - Clark Kimberling and John W. Layman, Mar 19 2011, corrected and edited by M. F. Hasler, Oct 12 2017
By definition, the array is symmetric, so row n = column n. Row 1 is essentially the same as A188434: T(n,1) = A101868(n) = A188434(n+1). - M. F. Hasler, Oct 12 2017
This product is commutative but is not associative and does not distribute over addition. - Peter Bala, Aug 13 2022

			   5 10 13 18  23 ...
  10 20 26 36  46
  13 26 34 47  60
  18 36 47 65  83
  23 46 60 83 106
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 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]

Cf. A101858, A101330, A101385, A101633 for similarly defined products.
Main diagonal is A101867.
First 3 rows are A101868, A101869, A101870.
Cf. A001622.

A101866[n_, k_] := n*k + Ceiling[n*GoldenRatio]*Ceiling[k*GoldenRatio];
Table[A101866[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

T(n, k) = my(phi = (1+sqrt(5))/2); n*k + ceil(phi*n)*ceil(phi*k); \\ Michel Marcus, Mar 29 2016

A101858 Array read by antidiagonals: T(n,k) = Porta-Stolarsky star product T(n,k) = n * k = nk + floor(phi n) floor(phi k) where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

2, 5, 5, 7, 13, 7, 10, 18, 18, 10, 13, 26, 25, 26, 13, 15, 34, 36, 36, 34, 15, 18, 39, 47, 52, 47, 39, 18, 20, 47, 54, 68, 68, 54, 47, 20, 23, 52, 65, 78, 89, 78, 65, 52, 23, 26, 60, 72, 94, 102, 102, 94, 72, 60, 26, 28, 68, 83, 104, 123, 117, 123, 104, 83, 68, 28, 31, 73, 94, 120

Offset: 1

N. J. A. Sloane, Jan 28 2005

			..2...5...7..10..13..15..18..20..23..26.
..5..13..18..26..34..39..47..52..60..68.
..7..18..25..36..47..54..65..72..83..94.
.10..26..36..52..68..78..94.104.120.136.
.13..34..47..68..89.102.123.136.157.178.
.15..39..54..78.102.117.141.156.180.204.
.18..47..65..94.123.141.170.188.217.246.
.20..52..72.104.136.156.188.208.240.272.
.23..60..83.120.157.180.217.240.277.314.
.26..68..94.136.178.204.246.272.314.356.

P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]

See A101330, A101385, A101633, A101866 for related definitions of product.
Main diagonal is A101863.
First 3 rows are A001950, A101864, A101865.
Cf. A001622.

A101858 := proc(n,k)
        phi := (1+sqrt(5))/2 ;
        n*k+floor(n*phi)*floor(phi*k) ;
end proc: # R. J. Mathar, Dec 06 2011

t[n_, k_] := n*k + Floor[n*GoldenRatio] * Floor[GoldenRatio*k]; Table[t[n-k, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".

Original entry on oeis.org

5, 8, 13, 18, 21, 26, 29, 34, 39, 42, 47, 52, 55, 60, 63, 68, 73, 76, 81, 84, 89, 94, 97, 102, 107, 110, 115, 118, 123, 128, 131, 136, 141, 144, 149, 152, 157, 162, 165, 170, 173, 178, 183, 186, 191, 196, 199, 204, 207, 212, 217, 220, 225, 228, 233, 238, 241, 246

Offset: 1

N. J. A. Sloane, Jan 26 2005

nonn

Numbers whose Zeckendorf representation ends with 000. - Benoit Cloitre, Jan 11 2014

The asymptotic density of this sequence is sqrt(5)-2. - Amiram Eldar, Mar 21 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
Donald E. Knuth, Fibonacci multiplication, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.

Second row of array in A101330.
Cf. A000201, A014417, A101642, A095085.
Set-wise difference of A026274 - A035337.

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]}]];
Table[kfp[2, n], {n, 60}] (* Robert G. Wilson v, Feb 04 2005 *)
With[{r = Map[Fibonacci, Range[2, 14]]}, Rest[-1 + Position[#, Integer][[All, 1]]] &@ Table[1/1000 * Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 0, 250}]] (* _Michael De Vlieger, Jun 08 2017 *)
Array[2*Floor[(#+1)*GoldenRatio]+#-2 &, 100] (* Paolo Xausa, Mar 20 2024 *)

from sympy import fibonacci
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return 1 if str(a(n))[-3:]=="000" else 0 # Indranil Ghosh, Jun 08 2017

from math import isqrt
def A101345(n): return (n+1+isqrt(5*(n+1)**2)&-2)+n-2 # Chai Wah Wu, Aug 29 2022

a(n) = floor(phi^3*(n+1)) - 3 - floor(2*phi*(n+1)) + 2*floor(phi*(n+1)) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Jan 11 2014

a(n) = 2*A000201(n+1) + n - 2. See the comments in A101642. - Michel Dekking, Dec 23 2019

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 04 2005

A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".

Original entry on oeis.org

8, 13, 21, 29, 34, 42, 47, 55, 63, 68, 76, 84, 89, 97, 102, 110, 118, 123, 131, 136, 144, 152, 157, 165, 173, 178, 186, 191, 199, 207, 212, 220, 228, 233, 241, 246, 254, 262, 267, 275, 280, 288, 296, 301, 309, 317, 322, 330, 335, 343, 351, 356, 364, 369, 377

Offset: 1

N. J. A. Sloane, Jan 26 2005

nonn

Let phi be the golden ratio. Using Fred Lunnon's formula in A101330 for Knuth's circle product, and the fact that phi^{-2} = 2-phi, plus [-x] = -[x]-1 for non-integer x, one obtains the formula below, expressing this sequence in terms of the lower Wythoff sequence. It follows in particular that the sequence of first differences 5,8,8,5,8,5,8,8,5,8,... of this sequence is the Fibonacci word A003849 on the alphabet {8,5}, shifted by 1. - Michel Dekking, Dec 23 2019
Also numbers with suffix string 0000, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 20 2024
The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 5.
W. F. Lunnon, Proof of formula, 2008.

Third row of array in A101330.
Cf. A101345 = Knuth's Fibonacci (or circle) product "2 o n".
Cf. A000201, A003849, A094214.

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[z1]}, {j, Length[z2]}]]; (* Robert G. Wilson v, Feb 04 2005 *)
Table[ kfp[3, n], {n, 50}] (* Robert G. Wilson v, Feb 04 2005 *)
Array[3*Floor[(# + 1)*GoldenRatio] + 2*# - 3 &, 100] (* Paolo Xausa, Mar 23 2024 *)

from math import isqrt
def A101642(n): return 3*(n+1+isqrt(5*(n+1)**2)>>1)+(n<<1)-3 # Chai Wah Wu, Aug 29 2022

From Michel Dekking, Dec 23 2019: (Start)
a(n) = 3*A000201(n+1) + 2n - 3.
a(n) = A101345(n) + A000201(n+1) + n + 1. (End)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 04 2005

A101332 a(n) = Knuth's Fibonacci (or circle) square "n o n".

Original entry on oeis.org

3, 8, 21, 40, 55, 84, 105, 144, 189, 220, 275, 336, 377, 448, 495, 576, 663, 720, 817, 880, 987, 1100, 1173, 1296, 1425, 1508, 1647, 1736, 1885, 2040, 2139, 2304, 2475, 2584, 2765, 2880, 3071, 3268, 3393, 3600, 3731, 3948, 4171, 4312, 4545, 4784, 4935

Offset: 1

N. J. A. Sloane, Jan 25 2005

T. D. Noe, Table of n, a(n) for n=1..1000

Main diagonal of A101330.

Array[3*#^2-2*#*Floor[(#+1)/GoldenRatio^2] &,100] (* Paolo Xausa, Mar 20 2024 *)

More terms from David Applegate, Jan 26 2005

More terms from T. D. Noe, May 07 2007

A135090 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 0, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 5, 5, 0, 0, 8, 8, 8, 0, 0, 11, 13, 13, 11, 0, 0, 13, 18, 21, 18, 13, 0, 0, 16, 21, 29, 29, 21, 16, 0, 0, 18, 26, 34, 40, 34, 26, 18, 0, 0, 21, 29, 42, 47, 47, 42, 29, 21, 0, 0, 24, 34, 47, 58, 55, 58, 47, 34, 24, 0, 0, 26, 39, 55, 65, 68, 68, 65, 55, 39, 26, 0, 0, 29, 42, 63, 76, 76, 84, 76, 76, 63, 42, 29, 0, 0, 32, 47

Offset: 0

N. J. A. Sloane, May 17 2008

This is a variant of A101330. See that entry for much more information.

			Array begins:
  n\k |  0   1   2   3   4    5    6    7    8    9 ...
  ----+------------------------------------------------
   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 ...
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Cf. A101330, A060144, A001622.

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 = 0..9)), n = 0..7);  # Peter Luschny, Mar 21 2024

A135090[n_, k_] := 3*n*k - n*Floor[(k + 1) / GoldenRatio^2] - k*Floor[(n + 1) / GoldenRatio^2];
Table[A135090[n-k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Mar 21 2024 *)

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

A340429 Array T(n, k) is the number x such that frac(xphi) + frac(nphi)frac(kphi) = 1 where phi is the golden ratio A001622 and frac(y) is the fractional part of y, read by antidiagonals.

Original entry on oeis.org

1, 3, 3, 4, 8, 4, 6, 11, 11, 6, 8, 16, 15, 16, 8, 9, 21, 22, 22, 21, 9, 11, 24, 29, 32, 29, 24, 11, 12, 29, 33, 42, 42, 33, 29, 12, 14, 32, 40, 48, 55, 48, 40, 32, 14, 16, 37, 44, 58, 63, 63, 58, 44, 37, 16, 17, 42, 51, 64, 76, 72, 76, 64, 51, 42, 17

Offset: 1

Michel Marcus, Jan 07 2021

			Array begins:
  1  3  4  6  8 ...
  3  8 11 16 21 ...
  4 11 15 22 29 ...
  6 16 22 32 42 ...
  8 21 29 42 55 ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Cf. A001622, A101330, A189663, A339765.

Cf. A000201 (row 1), A003623 (row 2), A190509 (row 3), A371388 (main diagonal).

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

A340429[n_, k_] := Floor[n * GoldenRatio] * k + Floor[k * GoldenRatio] * n - n * k;
Table[A340429[n - k + 1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 21 2024 *)

f(n) = 2*floor(n*(1+sqrt(5))/2) - 3*n; \\ A339765
T(n, k) = 2*n*k + f(n)*k/2 + f(k)*n/2;

T(n, k) = 2*n*k + A339765(n)*k/2 + A339765(k)*n/2.
T(n, k) = T(k, n), array is symmetric.
T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = ceiling(2*n / (sqrt(5) + 3)) = A189663(n + 1). - Peter Luschny, Mar 21 2024

A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.

Original entry on oeis.org

0, 1, 1, 2, 15, 2, 3, 8, 8, 3, 4, 12, 4, 12, 4, 5, 44, 18, 18, 44, 5, 6, 19, 24, 27, 24, 19, 6, 7, 62, 28, 96, 96, 28, 62, 7, 8, 26, 34, 42, 128, 42, 34, 26, 8, 9, 30, 14, 51, 56, 56, 51, 14, 30, 9, 10, 91, 44, 57, 180, 65, 180, 57, 44, 91, 10, 11, 37, 50, 66, 76, 79, 79, 76, 66, 50, 37, 11

Offset: 0

Peter Munn, Sep 11 2022

Square array A(x,y), x >= 0, y >= 0, defined as follows:
(1) Extend the Wythoff array infinitely to the left, maintaining the Fibonacci recurrence (see A287870 examples). We denote this extended array as eW(n,m), n >= 0, m any integer, indexed such that eW(n,0) = n. From each row n, form the set of pairs S_n = {(eW(n,m+1),eW(n,m)) : integer m)}.
(2) Define addition and multiplication of pairs by (j1,k1) + (j2,k2) = (j1+j2, k1+k2) and (j1,k1) o (j2,k2) = (j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2). (This defines a commutative ring with identity (1,0).)
(3) For nonnegative integers x and y, there is an integer z such that for every pair (j_x,k_x) in S_x and every pair (j_y,k_y) in S_y, (j_x,k_x) o (j_y,k_y) is in S_z. Define A(x,y) = z.
As a binary operation, A(.,.) is analogous to multiplication of coefficients in scientific numeric notation. The column position, m, used to define a pair in (1) above does not affect the eventual outcome, A(x,y), in (3), as no special pairs are selected from the pairs in S_x or S_y. The column position is analogous to the exponent. Notice also A(1,1) = 15 is substantially larger than A(2,2) = 4. This can be seen as analogous to 0.3 * 0.4 = 0.12 requiring more digits than 0.5 * 0.8 = 0.4.

			Calculation for A(1,2). Rows 1 and 2 of A287870 (indexed from 0) start 1, 3, ... and 2, 4, ... . So we may use the pairs (3,1) and (4,2). The defined multiplication gives (3*4 + 1*2, 3*2 + 4*1 - 1*2) = (14,8). 8, 14 , ... is in row 8 of A287870, so A(1,2) = 8.
For A(1,1), we start as above to get (3*3 + 1*1, 3*1 + 3*1 - 1*1) = (10,5). In the more general case, we form a sequence using the Fibonacci recurrence (as ..., 5, 10, ... may be in the extension leftwards of A287870). This starts 5, 10, 5+10=15, 10+15=25, 15+25=40, ... . We observe 15, 25, 40, ... is in row 15. So A(1,1) = 15.
The top left corner of the array:
  0   1   2    3    4    5    6    7    8    9
  1  15   8   12   44   19   62   26   30   91
  2   8   4   18   24   28   34   14   44   50
  3  12  18   27   96   42   51   57   66  198
  4  44  24   96  128   56  180   76   88  264
  5  19  28   42   56   65   79   33  102  116
  6  62  34   51  180   79  253  107  124  371
  7  26  14   57   76   33  107   45  138  157
  8  30  44   66   88  102  124  138  160  182
  9  91  50  198  264  116  371  157  182  544

Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.

Cf. A022344 (norm), A035513, A120873, A287870, A348853.

See the formula section for the relationships with A000201, A003622, A019586, A035336, A101330.

lowerw(n) = (n+sqrtint(5*n^2))\2 ; \\ A000201
upperw(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
compoundw(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ A003622
wpair(p) = {my(x=p[2], y = p[1], z); while(1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = upperw(n)); if ((x == xx) && (y == yy), return([xx, yy])); if (xx > x, ok = 0); n++;); z = y; y += x; x = z;);}
row(p) = {my(x=p[1], y=p[2], u); while (1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = compoundw(n)); if ((x == xx) && (y == yy), return(n)); if (xx > x, ok = 0); n++;); u = x; x = y - u; y = u;);} \\ similar to A120873
wrow(p) = row(wpair(p));
prodpair(v1, v2) = my(j1=v1[1], j2 = v2[1], k1 = v1[2], k2 = v2[2]); [j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2];
pair(n) = [lowerw(n+1), n];
T(n, k) = my(pn = pair(n), pk = pair(k), px = prodpair(pn, pk)); wrow(px)-1; \\ Michel Marcus, Sep 18 2022

A(x,y) = g(j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2), where j1 = A035336(x+1), j2 = A035336(y+1), k1 = A003622(x+1), k2 = A003622(y+1) and g(j,k) = (if j = A000201(k+1) then k otherwise g(k,j-k)).
A(x,y) = A(y,x).
A(x,0) = x.
A(x, A(y,z)) = A(A(x,y), z).
A022344(A(x,y)) = A022344(x) * A022344(y).
A(A019586(x), A019586(y)) = A019586(A101330(x,y)). (conjectured)

cp's OEIS Frontend

A101385 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330).

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A101646 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330). Sometimes called the "arroba" product.

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A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

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A101858 Array read by antidiagonals: T(n,k) = Porta-Stolarsky star product T(n,k) = n * k = nk + floor(phi n) floor(phi k) where phi = (1 + sqrt(5))/2.

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A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".

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A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".

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A101332 a(n) = Knuth's Fibonacci (or circle) square "n o n".

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A135090 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 0, k >= 0.

A340429 Array T(n, k) is the number x such that frac(xphi) + frac(nphi)frac(kphi) = 1 where phi is the golden ratio A001622 and frac(y) is the fractional part of y, read by antidiagonals.