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.

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A339950 Numbers k such that all k-sections of the infinite Fibonacci word A014675 have just two different run-lengths.

Original entry on oeis.org

1, 7, 14, 20, 27, 35, 41, 48, 54, 62, 69, 75, 82, 90, 96, 103, 109, 117, 124, 130, 137, 143, 151, 158, 164, 171, 179, 185, 192, 198, 206, 213, 219, 226, 234, 240, 247, 253, 260, 268, 274, 281, 287, 295, 302, 308, 315, 323, 329, 336, 342, 350, 357, 363, 370, 376, 384, 391, 397, 404
Offset: 1

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Author

Clark Kimberling, Dec 26 2020

Keywords

Comments

Equivalent definition: these are the numbers n such that all n-sections of the infinite Fibonacci word A003849 have just two run-lengths.
The distinct terms of the difference sequence of the first 40 terms are 6, 7, and 8.
Conjecture: a(n) = A189378(n-1)+1 for n >= 2. - Don Reble, Apr 06 2021.
"All n-sections" means all subsequences S(k) = (A014675(n*i+k); i = 0, 1, 2, ...), for k = 0, ..., n-1. "Run-lengths" means the numbers of consecutive equal terms in the sequence: see examples. - M. F. Hasler, Apr 07 2021

Examples

			Let W = A014675, so that as a word, W = 21221212212212122121221221212212212122121221221...
The unique 1-section of W is W itself, which is a concatenation of runs 1, 2, and 22, so that a(1) = 2. The sequence A339949 shows that a(n) > 2 for n = 2,3,4,5,6. For n = 7, the n-section of W that starts with its first letter, 2, is 221221221221221221221221221221221221121..., in which the runs are 22, 1, 11, supporting the conjecture that a(2) = 7.
Some run-lengths may appear quite late. For example, when n = 68, the third run-length appears in the n-section S(k=0) only with the 2829th element, corresponding to the 192372-th element of the original sequence. - _M. F. Hasler_, Apr 07 2021

Links

Crossrefs

Cf. A001622, A003849, A014675, A339949.
See also A189377, A189378, A189379.

Programs

  • Mathematica
    r = (1 + Sqrt[5])/2; z = 80000;
f[n_] := Floor[(n + 1) r] - Floor[n r];  (* A014675 *)
t = Table[Max[Map[Length,
     Union[Split[Table [f[n d], {n, 0, Floor[z/d]}]]]]], {d, 1,
    400}, {n, 1, d}];
u = Map[Max, t]
Flatten[Position[u, 2]]  (* A339950 *)

Extensions

More terms from Don Reble, Apr 13 2021

A364648 Starting position of the first occurrence of the longest monochromatic arithmetic progression of difference n in the Fibonacci infinite word (A003849).

Original entry on oeis.org

2, 3, 20, 16, 11, 20, 0, 143, 2, 11, 54, 8, 32, 2, 11, 7, 70, 3, 7, 0, 986, 10, 3, 7, 16, 11, 2, 87, 376, 2, 3, 2, 21, 87, 2, 3, 7, 16, 3, 7, 0, 20, 23, 11, 20, 8, 11, 2, 11, 20, 36, 11, 7, 0, 6764, 31, 3, 376, 84, 11, 54, 0, 20, 2, 3, 2, 42, 87, 2, 3, 54, 304
Offset: 1

Views

Author

Gandhar Joshi, Jul 31 2023

Keywords

Comments

From Gandhar Joshi, Jan 25 2025: (Start)
F(n) is the n-th Fibonacci number.
Conjecture: for n>0,
1. a(F(2n))=F(4n)-1; a(F(2n+1))=F(2n+3)-2.
2. a(F(6n)/2)=F(6n+3)/2-1; a(F(6n-3)/2)=F(6n)/2-2. (End)

Examples

			For the difference n = 3, the longest monochromatic progression has length A339949(3) = 5 and thus defined by f(i)=f(i+3)=f(i+6)=f(i+9)=f(i+12), where f(i) is the i-th term of the Fibonacci word (A003849); the smallest i for which that holds is i=20, so a(3) = 20.

Links

Crossrefs

Cf. A003849, A339949 (length of the longest monochromatic arithmetic progression).

Programs

  • Walnut
    # In the following line, replace every n with the desired constant difference, and every q with the longest MAP length for difference n given by A339949(n).
def f_n_map "?msd_fib Ak (k F[i]=F[i+n*k] & Aj (j ~(Ak (k F[j]=F[j+n*k])";
# Gandhar Joshi, Jan 25 2025

A370755 a(n) is the van der Waerden number W_f(2,n) of the Fibonacci word (A003849).

Original entry on oeis.org

1, 3, 8, 12, 21, 29, 42, 59, 67, 80, 88, 144, 160, 173, 186, 199, 220, 254, 377, 394, 423, 444, 465, 491, 512, 533, 554, 588, 609, 987, 1024, 1058, 1092, 1126, 1160, 1194, 1228, 1262, 1296, 1330, 1364, 1406, 1440, 1474, 1508, 1563, 1652, 2588, 2643, 2698, 2753
Offset: 1

Views

Author

Gandhar Joshi, Feb 29 2024

Keywords

Comments

a(n) is an extremely naive lower bound of the Waerden numbers A005346(n).

Examples

			For n=3, at least a(3)=8 terms of the prefix of the Fibonacci word are required to find a monochromatic arithmetic progression of length 3:
  Fibonacci word: 0 1 0 0 1 0 1 0 ...
                        ^   ^   ^
The 3 terms have equal values and are at locations which are a constant step apart (2 in this case).

References

  • B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw. Arch. Wisk. (in German), 15 (1927), 212-216.

Links

Crossrefs

Cf. A003849, A005346, A339949 (longest progression lengths), A364648 (first positions of longest progressions of length A339949(n)).

Programs

  • C
    /* See links. */
  • Walnut
    // The program is written for a fixed value of progression length, so it is run to find each a(n) separately. Following is an example to find a(5).
def fibw5map "?msd_fib F[i]=F[i+d] & F[i]=F[i+2*d] & F[i]=F[i+3*d] & F[i]=F[i+4*d]";
// This asserts that there is a progression of length 5 for difference d and first position i taken in pair.
def fibw5mapnew "?msd_fib $fibw5map(d,i) & d>0 & i+4*dA339949.
test fibw5mapnew 5;
// This enumerates the first 5 accepted pairs (d,i) in Zeckendorf representation listed in lexicographic order. The first or second in the list is our improved bound to be replaced for N in line number 2.
def fibw5mapfin "?msd_fib Ed,i ($fibw5map(d,i) & d>0 & i+4*d
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