A035336 -id:A035336 - cp's OEIS frontend

A134863 Wythoff BAB numbers.

7, 20, 28, 41, 54, 62, 75, 83, 96, 109, 117, 130, 143, 151, 164, 172, 185, 198, 206, 219, 227, 240, 253, 261, 274, 287, 295, 308, 316, 329, 342, 350, 363, 376, 384, 397, 405, 418, 431, 439, 452, 460, 473, 486, 494, 507, 520, 528, 541, 549, 562, 575, 583, 596

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAB = 2A+3B-1.
Also numbers with suffix string 1010, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 24 2024
The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - 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.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134864, A035513, A244593.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := B[A[B[n]]]; Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

from sympy import floor
from mpmath import phi
def A(n): return floor(n*phi)
def B(n): return floor(n*phi**2)
def a(n): return B(A(B(n))) # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A134863(n): return 5*(n+isqrt(5*n**2)>>1)+3*n-1 # Chai Wah Wu, Aug 11 2022

a(n) = B(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.
From A.H.M. Smeets, Mar 24 2024: (Start)
a(n) = 2*A(n) + 3*B(n) - 1 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.
Equals {A035336}\{A134861} (= Wythoff BA \ Wythoff BAA). (End)

A123740 Characteristic sequence for Wythoff AB-numbers A003623.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Wolfdieter Lang, Oct 13 2006

Left shifted sequence is the characteristic function of A035336, and also the second lowest digit of the Zeckendorf expansion of n. - Franklin T. Adams-Watters, Jun 30 2009
a(n) = A188009(n+2), n>=1. - Wolfdieter Lang, Jun 27 2011
Doubling the 0’s in the infinite Fibonacci word A003849 gives (a(n)). - Michel Dekking, Sep 09 2016
This is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism. The fixed point is the unique fixed point A270788 of the three symbol Fibonacci morphism. The letter-to-letter map is 1->0, 2->0, 3->1. - Michel Dekking, May 02 2019

See references under A000201.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [math.DS], 2016.
Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, Theoretical Computer Science 668 (2017), 59-69.
Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 31.

Cf. A003623, A000201, A001950, A188009, A270788.

Cf. A003849, A014417. - Franklin T. Adams-Watters, Jun 30 2009

a[] = 0; s = Table[n + 2 Floor[n*GoldenRatio], {n, 24}]; Map[Set[a[#], 1] &, s]; Array[a, Max[s]] (* _Michael De Vlieger, Mar 29 2023 *)

from math import isqrt
def A123740(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+isqrt(m-20*(n+1))>>1)-3 # Chai Wah Wu, Aug 29 2022

a(n) = 1 if n=A(B(k)) for some k>=1, else 0, with A(k):=A000201(k) and B(k):=A001950(k), k>=1.
a(n) = 1-(1-h(n))-(1-h(n+1)) = h(n)-(1-h(n+1))= h(n)*h(n+1) with h(n):=A005614(n-1), n>=1, the rabbit sequence.
a(n) = A(n+2)-A(n)-3. - Wolfdieter Lang, Jun 27 2011

A005713 Define strings S(0)=0, S(1)=11, S(n) = S(n-1)S(n-2); iterate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane

a(A035336(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = 1 - A123740(n).  This can be seen as follows. Define words T(0)=0, T(1)=1, T(n) = T(n-1)T(n-2). Then T(infinity) is the binary complement of the infinite Fibonacci word A003849. Obviously S(n) is the [1->11] transform of T(n). The claim now follows from the observation (see Comments of A123740) that doubling the 0's in the infinite Fibonacci word A003849 gives A123740. - Michel Dekking, Oct 21 2018
From Michel Dekking, Oct 22 2018: (Start)
Here is a proof of Cloitre's (corrected) formula
      a(n) = abs(A014677(n+1)).
Since abs(-1) = abs(1) = 1, one has to prove that A014677(k)=0 if and only if there is an n such that AB(n) = k (using that a(n) = 1 - A123740(n)). Now A014677 is the sequences of first differences of A001468, and the 0's in A014677 occur if and only if there occurs a block 22 in A001468, which is given by
      A001468(n) = floor((n+1)*phi) - floor(n*phi), n >= 0.
But then
      A001468(n) = A014675(n-1), n > 0.
The sequence A014675 is fixed point of the morphism 1->2, 2->21, which is alphabet equivalent to the morphism 1->12, 2->1, the classical Fibonacci morphism in standard form. This implies that the 22 blocks in A001468 occur at position n+1 in if and only if 3 occurs in the fixed point A270788 of the 3-symbol Fibonacci morphism at k, which happens if and only if there is an n such that AB(n)=k (see Formula of A270788). (End)

			The infinite word is S(infinity) = 110111101101111011110110...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
R. K. Guy, Letter to N. J. A. Sloane, 1987
Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 31.

Cf. A005614, A003849.

Cf. A001468, A014675, A014677, A123740, A270788.

a005713 n = a005713_list !! n
a005713_list = 1 : 1 : concat (sibb [0] [1,1]) where
   sibb xs ys = zs : sibb ys zs where zs = xs ++ ys
-- Reinhard Zumkeller, Dec 30 2011

s[0] = {0}; s[1] = {1, 1}; s[n_] := s[n] = Join[s[n-1], s[n-2]]; s[10] (* Jean-François Alcover, May 15 2013 *)
nxt[{a_,b_}]:={b,Join[a,b]}; Drop[Nest[nxt,{{0},{1,1}},10][[1]],3] (* Harvey P. Dale, Jan 31 2019 *)

a(n,f1,f2)=local(f3); for(i=3,n,f3=concat(f2,f1); f1=f2; f2=f3); f2

printp(a(10,[ 0 ],[ 1,1 ])) \\ Would give S(10). Sequence is S(infinity).

From Benoit Cloitre, Apr 21 2003: (Start)
For n > 1, a(n-1) = floor(phi*ceiling(n/phi)) - ceiling(phi*floor(n/phi)) where phi=(1+sqrt(5))/2.
For n >= 0, a(n) = abs(A014677(n+1)). (End)

Corrected by Michael Somos

A035339 5th column of Wythoff array.

Original entry on oeis.org

8, 29, 42, 63, 84, 97, 118, 131, 152, 173, 186, 207, 228, 241, 262, 275, 296, 317, 330, 351, 364, 385, 406, 419, 440, 461, 474, 495, 508, 529, 550, 563, 584, 605, 618, 639, 652, 673, 694, 707, 728, 741, 762, 783, 796, 817, 838, 851, 872, 885, 906, 927, 940, 961

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), this sequence (k=5), A035340 (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(8*floor((n+1)*t)+5*n,n=0..80) ];

a[n_] := 8 * Floor[n * GoldenRatio] + 5*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A035340 6th column of Wythoff array.

Original entry on oeis.org

13, 47, 68, 102, 136, 157, 191, 212, 246, 280, 301, 335, 369, 390, 424, 445, 479, 513, 534, 568, 589, 623, 657, 678, 712, 746, 767, 801, 822, 856, 890, 911, 945, 979, 1000, 1034, 1055, 1089, 1123, 1144, 1178, 1199, 1233, 1267, 1288, 1322, 1356, 1377, 1411, 1432

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^7 = A094214^7 = 0.03444185... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), A035339 (k=5), this sequence (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(13*floor((n+1)*t)+8*n,n=0..80) ];

a[n_] := 13 * Floor[n * GoldenRatio] + 8*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A276885 Sums-complement of the Beatty sequence for 1 + phi.

Original entry on oeis.org

1, 4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237

Offset: 1

Clark Kimberling, Oct 01 2016

See A276871 for a definition of sums-complement and guide to related sequences.
This appears to be 1 followed by A089910. - R. J. Mathar, Oct 05 2016
Mathar's conjecture is proved in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'. See Example 1 in that paper. - Michel Dekking, Dec 21 2017

			The Beatty sequence for 1 + phi is A001950 = (2,5,7,10,13,15,18,20,23,26,...), with difference sequence s = A005614 + 2 = (3,2,3,3,2,3,2,3,3,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,7,8,10,11,13,14,15,16,18,...), with complement (1,4,9,12,17,22,...).

Jeffrey Shallit, "Synchronized Sequences" in Lecture Notes of Computer science 12847 pp 1-19 2021, see page 16.

Michel Dekking, The Frobenius problem for homomorphic embeddings of languages into the integers, arXiv:1712.03345 [math.CO], 2017.
Index entries for sequences related to Beatty sequences

Cf. A001950, A003849, A276871.

z = 500; r = 1 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A001950 *)
t = Differences[b]; (* 2 + A003849 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]; (* A276885 *)

from math import isqrt
def A276885(n): return n+(n-1+isqrt(5*(n-1)**2)&-2) # Chai Wah Wu, May 21 2025

a(n) = 2[(n-1)phi] + n, where phi = (1+sqrt(5))/2 (see Example 1 in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'). - Michel Dekking, Dec 21 2017

a(n) = A035336(n-1)+2 for n>1. - Michel Dekking, Dec 21 2017

Name edited and example corrected by Michel Dekking, Oct 30 2016

A134571 Array T(n,k) by antidiagonals; T(n,k) = position in row n of k-th occurrence of the Fibonacci number F(2n) in A134567.

Original entry on oeis.org

1, 3, 2, 4, 7, 5, 6, 10, 18, 13, 8, 15, 26, 47, 34, 9, 20, 39, 68, 123, 89, 11, 23, 52, 102, 178, 322, 233

Offset: 1

Clark Kimberling, Nov 02 2007

(Row 1) = A000201, the lower Wythoff sequence (Row 2) = (Column 2 of Wythoff array) = A035336 (Row 3) = (Column 4 of Wythoff array) = A035338 (Row 4) = (Column 6 of Wythoff array) = A035340 (Column 1) = A001519 (bisection of Fibonacci sequence) (Column 2) = A005248 (bisection of Lucas sequence) (Column 3) = A052995 Row 1 is the ordered union of all odd-numbered columns of the Wythoff array; and A134571 is a permutation of the positive integers.

It looks like this array is A080164 transposed. - Peter Munn, Sep 02 2025

			Northwest corner:
1 3 4 6 8 9 11 12 14 16
2 7 10 15 20 23
5 18 26 39 52 60
13 47 68 102 136 157
Row 1 consists of numbers k such that 1 is the least m for which {-m*tau}<{k*tau}, where tau=(1+sqrt(5))/2 and {} denotes fractional part.

Cf. A080164, A134567, A134570.

A219639 Numbers that occur only once in A219641.

Original entry on oeis.org

1, 5, 8, 13, 17, 21, 25, 28, 34, 38, 41, 46, 50, 55, 59, 62, 67, 71, 75, 79, 82, 89, 93, 96, 101, 105, 109, 113, 116, 122, 126, 129, 134, 138, 144, 148, 151, 156, 160, 164, 168, 171, 177, 181, 184, 189, 193, 198, 202, 205, 210, 214, 218, 222, 225, 233, 237, 240

Offset: 1

Antti Karttunen, Nov 24 2012

nonn

Fibonacci numbers, A000045(i) from i>=5 onward, 5, 8, 13, 21, ..., all occur in this sequence, as well as these numbers plus 2: 7, 10, 15, 23, all have Zeckendorf-expansions with just two 1's and end with ...010.

A. Karttunen, Table of n, a(n) for n = 1..10000

Cf. A219637.

a(n) = A219641(A035336(n)).

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)

A095087 Fib010 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero, one and zero.

Original entry on oeis.org

7, 23, 31, 41, 83, 109, 151, 167, 193, 227, 269, 277, 311, 337, 353, 379, 397, 421, 431, 439, 463, 523, 541, 557, 599, 607, 617, 641, 659, 701, 709, 719, 727, 743, 761, 769, 811, 829, 853, 863, 887, 929, 947, 997, 1031, 1049, 1091, 1117, 1151

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A035336. Cf. A095067.

from sympy import fibonacci, primerange
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 str(a(n))[-3:]=="010"
print([n for n in primerange(1, 1201) if ok(n)]) # Indranil Ghosh, Jun 08 2017

cp's OEIS Frontend

A134863 Wythoff BAB numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

A123740 Characteristic sequence for Wythoff AB-numbers A003623.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A005713 Define strings S(0)=0, S(1)=11, S(n) = S(n-1)S(n-2); iterate.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A035339 5th column of Wythoff array.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A035340 6th column of Wythoff array.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A276885 Sums-complement of the Beatty sequence for 1 + phi.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A134571 Array T(n,k) by antidiagonals; T(n,k) = position in row n of k-th occurrence of the Fibonacci number F(2n) in A134567.

Views

Author

Keywords

Comments

Examples