A047924 -id:A047924 - cp's OEIS frontend

A038150 Array of numbers used in exotic ternary numeration system, read by antidiagonals.

1, 2, 3, 4, 6, 8, 5, 11, 16, 21, 7, 14, 29, 42, 55, 9, 19, 37, 76, 110, 144, 10, 24, 50, 97, 199, 288, 377, 12, 27, 63, 131, 254, 521, 754, 987, 13, 32, 71, 165, 343, 665, 1364, 1974, 2584, 15, 35, 84, 186, 432, 898, 1741, 3571, 5168, 6765, 17, 40, 92, 220, 487

Offset: 0

Aviezri S. Fraenkel

			Top left corner of array:
  1,  3,  8, 21,  55, ...
  2,  6, 16, 42, 110, ...
  4, 11, 29, 76, 199, ...
  5, 14, 37, 97, 254, ...

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
A. S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.

Rows give A001906, A025169, A002878.
Columns give A026351, A047924, A047925.
Main diagonal gives A047923.
Cf. A026351, A035506.

t[n_, 1] := Floor[(n - 1) GoldenRatio] + 1; t[n_, j_] := Floor[ GoldenRatio^2 t[n, j - 1]] + 1; Table[ t[n - m + 1, m], {n, 11}, {m, n}] // Flatten (* Birkas Gyorgy, Apr 15 2011; modified by Robert G. Wilson v, Apr 15 2011 *)

For n >= 0, A_0^n is the least nonnegative integer not in {A_j^n: 0 <= i < n, j >= 0, A_1^n = 2A_0^n + n, A_j^n = 3A_{j-1}^n - A_{j-2}^n (j >= 2).

a(n,k) = F(2k)*n + F(2k+1)*A026351(n). - Charlie Neder, Feb 07 2019

More terms from Naohiro Nomoto, Jun 07 2001

A223025 Gives the column number which contains n in the dual Wythoff array (beginning the column count at 1).

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 1, 3, 2, 1, 6, 2, 1, 4, 1, 3, 2, 1, 7, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 8, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 9, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 7, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2

Offset: 1

Casey Mongoven, Mar 11 2013

nonn

			a(23) = 3 because 23 is in the third column of the dual Wythoff array (see A126714).

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A007066, A126714, A047924, A167198, A035614.

A383671 The limiting word that starts with 0, as a sequence, generated by s(0) = 0, s(1) = 12, s(n) = concatenation of s(n - 2) and s(n - 1).

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2

Offset: 0

Clark Kimberling, May 15 2025

nonn

There are two distinct limiting words generated by s(0) = 0, s(1) = 12, s(n) = s(n - 2)s(n - 1). This one is given by s(2n) for n>=0; the other, given by s(2n-1) for n>=0, is 1201201212012... In both limiting words, the length of the n-th initial subword is A000045(n+1), for n>=1.

			Initial subwords: s(0)=0, s(1)=12, s(2)=012, s(3)=12012, s(4)= 01212012, of lengths 1, 2, 3, 5, 8 (Fibonacci numbers).

Cf. A000045, A003849, A047924 (positions of 0), A026356 (positions of 1), A022413 (positions of 2), A383670.

s[0] = "0"; s[1] = "12"; s[n_] := StringJoin[s[n - 2], s[n - 1]];
Join[{0}, IntegerDigits[FromDigits[s[10]]]]

from math import isqrt
def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1
def A026356(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n
def A383671(n):
    def bsearch(f, n):
        kmin, kmax = 0, 1
        while f(kmax) <= n:
            kmax <<= 1
        kmin = kmax>>1
        while True:
            kmid = kmax+kmin>>1
            if f(kmid) > n:
                kmax = kmid
            else:
                kmin = kmid
            if kmax-kmin <= 1:
                break
        return kmin
    if n<3: return n
    for i, f in enumerate((A047924, A026356)):
        if f(bsearch(f,n+1))==n+1: return i
    return 2 # Chai Wah Wu, May 21 2025

cp's OEIS Frontend

A038150 Array of numbers used in exotic ternary numeration system, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A223025 Gives the column number which contains n in the dual Wythoff array (beginning the column count at 1).

Views

Author

Keywords

Examples

References

Links

Crossrefs

A383671 The limiting word that starts with 0, as a sequence, generated by s(0) = 0, s(1) = 12, s(n) = concatenation of s(n - 2) and s(n - 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python