A101385 -id:A101385 - cp's OEIS frontend

A101633 Main diagonal of array in A101385.

3, 34, 987, 1032, 75025, 75138, 76279, 14930352, 14930643, 14935554, 15024075, 15024408, 7778742049, 7778742806, 7778763975, 7779378658, 7779379457, 7797272004, 7797272871, 7797295150, 10610209857723, 10610209859700

Offset: 1

N. J. A. Sloane, Jan 26 2005

nonn

Cf. A101385.

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 *)
Table[ kfpv[n, n], {n, 18}] (* Robert G. Wilson v, Feb 09 2005 *)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 09 2005

A101643 First row of array in A101385.

Original entry on oeis.org

3, 8, 21, 24, 55, 58, 63, 144, 147, 152, 165, 168, 377, 380, 385, 398, 401, 432, 435, 440, 987, 990, 995, 1008, 1011, 1042, 1045, 1050, 1131, 1134, 1139, 1152, 1155, 2584, 2587, 2592, 2605, 2608, 2639, 2642, 2647, 2728, 2731, 2736, 2749, 2752, 2961, 2964

Offset: 1

N. J. A. Sloane, Jan 25 2005, Jan 27 2005

nonn

Cf. A101385.

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 *)
Table[ kfpv[n, 1], {n, 48}] (* Robert G. Wilson v, Feb 09 2005 *)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 09 2005

A101644 Second row of array in A101385.

Original entry on oeis.org

8, 34, 144, 152, 610, 618, 644, 2584, 2592, 2618, 2728, 2736, 10946, 10954, 10980, 11090, 11098, 11556, 11564, 11590, 46368, 46376, 46402, 46512, 46520, 46978, 46986, 47012, 48952, 48960, 48986, 49096, 49104, 196418, 196426, 196452, 196562

Offset: 1

N. J. A. Sloane, Jan 25 2005, Jan 27 2005

Cf. A101385.

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]}]];
Table[ kfpv[n, 2], {n, 40}] (* Robert G. Wilson v, Feb 09 2005 *)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 09 2005

A101645 Third row of array in A101385.

Original entry on oeis.org

21, 144, 987, 1008, 6765, 6786, 6909, 46368, 46389, 46512, 47355, 47376, 317811, 317832, 317955, 318798, 318819, 324576, 324597, 324720, 2178309, 2178330, 2178453, 2179296, 2179317, 2185074, 2185095, 2185218, 2224677, 2224698

Offset: 1

N. J. A. Sloane, Jan 25 2005, Jan 27 2005

Cf. A101385.

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 *)
Table[ kfpv[n, 3], {n, 30}] (* Robert G. Wilson v, Feb 09 2005 *)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 09 2005

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

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.

Original entry on oeis.org

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

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 *)

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

A219875 Multiplication table of the operation "n o m" = nm + ceiling(n/phi) ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

Original entry on oeis.org

2, 4, 4, 5, 8, 5, 7, 10, 10, 7, 9, 14, 13, 14, 9, 10, 18, 18, 18, 18, 10, 12, 20, 23, 25, 23, 20, 12, 13, 24, 26, 32, 32, 26, 24, 13, 15, 26, 31, 36, 41, 36, 31, 26, 15, 17, 30, 34, 43, 46, 46, 43, 34, 30, 17, 18, 34, 39, 47, 55, 52, 55, 47, 39, 34, 18

Offset: 1

Michel Marcus, Dec 01 2012

Like A101866, this operation is associative.
First rows of the table are:
1: 2,  4,  5,  7,  9, 10, 12,  13,  15,  17, ...
2: 4,  8, 10, 14, 18, 20, 24,  26,  30,  34, ...
3: 5, 10, 13, 18, 23, 26, 31,  34,  39,  44, ...
4: 7, 14, 18, 25, 32, 36, 43,  47,  54,  61, ...
5: 9, 18, 23, 32, 41, 46, 55,  60,  69,  78, ...
6:10, 20, 26, 36, 46, 52, 62,  68,  78,  88, ...
7:12, 24, 31, 43, 55, 62, 74,  81,  93, 105, ...
8:13, 26, 34, 47, 60, 68, 81,  89, 102, 115, ...
9:15, 30, 39, 54, 69, 78, 93, 102, 117, 132, ...
Row 1 is A004956.
Row 3 is A101868.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
P. Arnoux, Some remarks about Fibonacci multiplication, Applied Mathematics Letters, Volume 2, Issue 4, 1989, Pages 319-320.

Cf. A001622, A004956, A101385, A101858, A101866, A101868, A371381 (main diagonal).

A219875[n_, m_] := n*m + Ceiling[n / GoldenRatio] * Ceiling[m / GoldenRatio];
Table[A219875[n-m+1, m], {n, 15}, {m, n}] (* Paolo Xausa, Mar 20 2024 *)

prod(m,n) = {phi = (1+sqrt(5))/2; return (m*n + ceil(m/phi)*ceil(n/phi));}

cp's OEIS Frontend

A101633 Main diagonal of array in A101385.

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A101643 First row of array in A101385.

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A101644 Second row of array in A101385.

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A101645 Third row of array in A101385.

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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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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.

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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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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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A219875 Multiplication table of the operation "n o m" = n*m + ceiling(n/phi)* ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

A219875 Multiplication table of the operation "n o m" = nm + ceiling(n/phi) ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.