A101858 -id:A101858 - cp's OEIS frontend

A101863 Main diagonal of A101858.

2, 13, 25, 52, 89, 117, 170, 208, 277, 356, 410, 505, 610, 680, 801, 881, 1018, 1165, 1261, 1424, 1530, 1709, 1898, 2020, 2225, 2440, 2578, 2809, 2957, 3204, 3461, 3625, 3898, 4181, 4361, 4660, 4850, 5165, 5490, 5696, 6037, 6253, 6610, 6977, 7209, 7592, 7985

Offset: 1

N. J. A. Sloane, Jan 28 2005

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001622, A101858.

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

a(n) = n^2 + floor(n*(1 + sqrt(5))/2)^2. - Paolo Xausa, Mar 20 2024

A101865 Third row of A101858.

Original entry on oeis.org

7, 18, 25, 36, 47, 54, 65, 72, 83, 94, 101, 112, 123, 130, 141, 148, 159, 170, 177, 188, 195, 206, 217, 224, 235, 246, 253, 264, 271, 282, 293, 300, 311, 322, 329, 340, 347, 358, 369, 376, 387, 394, 405, 416, 423, 434, 445, 452, 463, 470, 481, 492, 499, 510, 517, 528, 539

Offset: 1

N. J. A. Sloane, Jan 28 2005

nonn

A101864 Wythoff BB numbers.

Original entry on oeis.org

5, 13, 18, 26, 34, 39, 47, 52, 60, 68, 73, 81, 89, 94, 102, 107, 115, 123, 128, 136, 141, 149, 157, 162, 170, 178, 183, 191, 196, 204, 212, 217, 225, 233, 238, 246, 251, 259, 267, 272, 280, 285, 293, 301, 306, 314, 322, 327, 335, 340, 348, 356, 361, 369, 374, 382, 390, 395

Offset: 1

N. J. A. Sloane, Jan 28 2005

nonn

a(n)-3 are also the positions of 1 in A188436. - Federico Provvedi, Nov 22 2018

The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Jean-Paul Allouche and F. Michel Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018-2019.
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.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008), Article 08.3.3.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik, Vol. 78, No. 2 (2021), pp. 1-8.
Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), 267-273.

Second row of A101858.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
Cf. A094214, A188436.

b:=n->floor(n*((1+sqrt(5))/2)^2): seq(b(b(n)),n=1..60); # Muniru A Asiru, Dec 05 2018

b[n_] := Floor[n * GoldenRatio^2]; a[n_] := b[b[n]]; Array[a, 60] (* Amiram Eldar, Nov 22 2018 *)

from sympy import S
for n in range(1,60): print(int(S.GoldenRatio**2*(int(n*S.GoldenRatio**2))), end=', ') # Stefano Spezia, Dec 06 2018

a(n) = B(B(n)), n>=1, with B(k)=A001950(k) (Wythoff B-numbers). a(0)=0 with B(0)=0.

A033891 a(n) = Fibonacci(4*n+3).

Original entry on oeis.org

2, 13, 89, 610, 4181, 28657, 196418, 1346269, 9227465, 63245986, 433494437, 2971215073, 20365011074, 139583862445, 956722026041, 6557470319842, 44945570212853, 308061521170129, 2111485077978050

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).

Cf. A000045, A004187, A049685, A101858, A167816.

List([0..30], n-> Fibonacci(4*n+3)); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+3): n in [0..30]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[4*n+3],{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)
LinearRecurrence[{7,-1},{2,13},31] (* or *) CoefficientList[Series[ (2-x)/(1-7x+x^2),{x,0,30}],x] (* Harvey P. Dale, May 03 2011 *)

a(n)=fibonacci(4*n+3) \\ Charles R Greathouse IV, Sep 24 2015

[fibonacci(4*n+3) for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = 7*a(n-1) - a(n-2). - Floor van Lamoen, Dec 10 2001
G.f.: (2-x)/(1-7*x+x^2). - Philippe Deléham, Nov 17 2008
a(n) = A167816(4*n+3). - Reinhard Zumkeller, Nov 13 2009
a(n) = Fibonacci(2*n+2)^2 + Fibonacci(2*n+1)^2. - Gary Detlefs, Oct 12 2011
a(n) = 2*A004187(n+1) - A004187(n). - R. J. Mathar, Nov 26 2011
a(n) = A004187(n+1) + A049685(n). - Yuriy Sibirmovsky, Sep 15 2016
From  Peter Bala, Aug 11 2022: (Start)
Let n ** m =  n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 2 ** 2 ** ... ** 2 (n+1 factors).
a(2*n+1) = a(n) ** a(n) = Fibonacci(8*n+7); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(12*n+11) and so on. (End)

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

A134497 a(n) = Fibonacci(6n+5).

Original entry on oeis.org

5, 89, 1597, 28657, 514229, 9227465, 165580141, 2971215073, 53316291173, 956722026041, 17167680177565, 308061521170129, 5527939700884757, 99194853094755497, 1779979416004714189, 31940434634990099905, 573147844013817084101, 10284720757613717413913

Offset: 0

Artur Jasinski, Oct 28 2007

Colin Barker, Table of n, a(n) for n = 0..750
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A049660, A101858, A103134, A134492, A134493, A134494, A134495.

[Fibonacci(6*n +5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n+5], {n, 0, 30}]
Take[Fibonacci[Range[100]],{5,-1,6}] (* Harvey P. Dale, Jun 18 2013 *)

a(n)=fibonacci(6*n+5) \\ Charles R Greathouse IV, Jun 11 2015

Vec((5-x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

G.f.: ( 5-x ) / ( 1-18*x+x^2 ). a(n) = 5*A049660(n+1)-A049660(n). - R. J. Mathar, Apr 17 2011
a(n) = A000045(A016969(n)). - Michel Marcus, Nov 08 2013
a(n) = ((25-11*sqrt(5)+(9+4*sqrt(5))^(2*n)*(25+11*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = 5*S(n, 18) - S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018
From Peter Bala, Aug 11 2022: (Start)
Let n ** m =  n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 5 ** 5 ** ... ** 5 (n+1 factors).
a(2*n+1) = a(n) ** a(n) = Fibonacci(12*n+11); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(18*n+17) and so on. (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

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));}

A295573 Array read by upwards antidiagonals: T(n,k) = nk + floor(phi n) ceiling(phi k) where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

3, 8, 6, 11, 16, 8, 16, 22, 21, 11, 21, 32, 29, 29, 14, 24, 42, 42, 40, 37, 16, 29, 48, 55, 58, 51, 42, 19, 32, 58, 63, 76, 74, 58, 50, 21, 37, 64, 76, 87, 97, 84, 69, 55, 24, 42, 74, 84, 105, 111, 110, 100, 76, 63, 27, 45, 84, 97, 116, 134, 126, 131, 110, 87, 71, 29, 50, 90, 110, 134, 148, 152, 150, 144, 126, 98, 76, 32

Offset: 1

N. J. A. Sloane, Dec 03 2017

This is a hybrid of the Porta-Stolarsky star product (A101858) and the Arnoux product (A101866)

			The array begins:
3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, ...
8, 16, 21, 29, 37, 42, 50, 55, 63, 71, 76, 84, ...
11, 22, 29, 40, 51, 58, 69, 76, 87, 98, 105, 116, ...
16, 32, 42, 58, 74, 84, 100, 110, 126, 142, 152, 168, ...
21, 42, 55, 76, 97, 110, 131, 144, 165, 186, 199, 220, ...
24, 48, 63, 87, 111, 126, 150, 165, 189, 213, 228, 252, ...
29, 58, 76, 105, 134, 152, 181, 199, 228, 257, 275, 304, ...
32, 64, 84, 116, 148, 168, 200, 220, 252, 284, 304, 336, ...
...

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

Cf. A001622, A101858, A101866, A371382 (main diagonal).

T := proc(n, k) local phi;
        phi := (1+sqrt(5))/2 ;
        n*k+floor(n*phi)*ceil(phi*k) ;
end proc:
for n from 1 to 12 do
lprint([seq(T(n-i+1,i),i=1..n)]);
od: # by antidiagonals
for n from 1 to 12 do
lprint([seq(T(n,i),i=1..12)]);
od: # by rows

A295573[n_, k_] := n*k + Floor[n * GoldenRatio] * Ceiling[k * GoldenRatio];
Table[A295573[n-k+1,k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

cp's OEIS Frontend

A101863 Main diagonal of A101858.

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A101865 Third row of A101858.

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A101864 Wythoff BB numbers.

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A033891 a(n) = Fibonacci(4*n+3).

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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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A134497 a(n) = Fibonacci(6n+5).

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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.

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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.