cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A134859 Wythoff AAA numbers.

Original entry on oeis.org

1, 6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242
Offset: 1

Views

Author

Clark Kimberling, Nov 14 2007

Keywords

Comments

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAA = AB - 2 and AAA = A + B - 2.
Also numbers with suffix string 001, when written in Zeckendorf representation (with leading zero for the first term). - A.H.M. Smeets, Mar 20 2024
The asymptotic density of this sequence is 1/phi^3 = phi^3 - 4 = A098317 - 4 = 0.236067... . - Amiram Eldar, Mar 24 2025

Examples

			Starting with A=(1,3,4,6,8,9,11,12,14,16,17,19,...), we have A(2)=3, so A(A(2))=4, so A(A(A(2)))=6.

Links

Crossrefs

Cf. A001622, A000201, A001950, A003622, A003623, A035336, A098317, A101864, A134860, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
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.
Essentially the same as A095098.

Programs

  • Maple
    # For Maple code for these Wythoff compound sequences see A003622. - N. J. A. Sloane, Mar 30 2016
  • Mathematica
    A[n_] := Floor[n GoldenRatio];
a[n_] := A@ A@ A@ n;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019 *)
  • Python
    from sympy import floor
from mpmath import phi
def A(n): return floor(n*phi)
def a(n): return A(A(A(n))) # Indranil Ghosh, Jun 10 2017
  • Python
    from math import isqrt
def A134859(n): return ((n+isqrt(5*n**2)>>1)-1<<1)+n # Chai Wah Wu, Aug 10 2022

Formula

a(n) = A(A(A(n))), n >= 1, with A=A000201, the lower Wythoff sequence.
a(n) = 2*floor(n*Phi^2) - n - 2 where Phi = (1+sqrt(5))/2. - Benoit Cloitre, Apr 12 2008; R. J. Mathar, Oct 16 2009
a(n) = A095098(n-1), n > 1. - R. J. Mathar, Oct 16 2009
From A.H.M. Smeets, Mar 23 2024: (Start)
a(n) = A(n) + B(n) - 2 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.
Equals {A003622}\{A134860} (= Wythoff AA \ Wythoff AAB). (End)

Extensions

Incorrect PARI program removed by R. J. Mathar, Oct 16 2009

A134860 Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.

Original entry on oeis.org

4, 12, 17, 25, 33, 38, 46, 51, 59, 67, 72, 80, 88, 93, 101, 106, 114, 122, 127, 135, 140, 148, 156, 161, 169, 177, 182, 190, 195, 203, 211, 216, 224, 232, 237, 245, 250, 258, 266, 271, 279, 284, 292, 300, 305, 313, 321, 326, 334, 339, 347, 355, 360, 368, 373
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004 and Clark Kimberling, Nov 14 2007

Keywords

Comments

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAB=AA+AB and AAB=A+2B-1.
The asymptotic density of this sequence is 1/phi^4 = 2/(7+3*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 21 2022

Links

Crossrefs

Cf. A000201, A001622, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
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.
Set-wise difference A003622 \ A095098. Cf. A095089 (fib101 primes).

Programs

  • Mathematica
    With[{r = Map[Fibonacci, Range[2, 14]]}, Position[#, {1, 0, 1}][[All, 1]] &@ Table[If[Length@ # < 3, {}, Take[#, -3]] &@ IntegerDigits@ Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &,Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 373}]] (* Michael De Vlieger, Jun 09 2017 *)
  • Python
    from sympy import fibonacci
def a(n):
    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:]=="101"
print([n for n in range(4, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017
  • Python
    from math import isqrt
def A134860(n): return 3*(n+isqrt(5*n**2)>>1)+(n<<1)-1 # Chai Wah Wu, Aug 10 2022

Formula

a(n) = A(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.

Extensions

This is the result of merging two sequences which were really the same. - N. J. A. Sloane, Jun 10 2017

A095086 Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.

Original entry on oeis.org

19, 43, 53, 61, 103, 137, 163, 179, 197, 229, 239, 263, 281, 307, 331, 349, 383, 433, 467, 509, 569, 577, 619, 653, 739, 773, 797, 823, 839, 857, 883, 907, 941, 967, 1009, 1051, 1061, 1069, 1103, 1129, 1153, 1171, 1187, 1213, 1229, 1289
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Links

Crossrefs

Intersection of A000040 and A095098. Cf. A014417, A095066.

Programs

  • Python
    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:]=="001"
print([n for n in primerange(1, 1301) if ok(n)]) # Indranil Ghosh, Jun 08 2017
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