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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A071160 Łukasiewicz words that are also valid asynchronic siteswap juggling patterns.

Original entry on oeis.org

0, 1, 20, 11, 300, 201, 120, 111, 4000, 3001, 2020, 2011, 1300, 1201, 1120, 1111, 50000, 40001, 30020, 30011, 20300, 20201, 20120, 20111, 14000, 13001, 12020, 12011, 11300, 11201, 11120, 11111, 600000, 500001, 400020, 400011, 300300
Offset: 0

Views

Author

Antti Karttunen, May 14 2002

Keywords

Comments

Note: this finite decimal representation works only up to the 511th term, as the 512th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071161 shows the initial portion of this sequence sorted.

Links

Crossrefs

Subset of A071153.
Cf. A060495, A060498, A065177, A071162, A071163.

Formula

Construction: starting from the most significant (the leftmost) bit, replace each 1-bit in the binary expansion of n with the distance to the next 1-bit to the right, allowing a cyclic wrap-over from the least-significant 1-bit to the most significant 1-bit. I.e. from 22 = 10110 in binary we get 20120, the 22nd term of this sequence.
a(n) = A071161(A054429(n)).

A071152 Łukasiewicz words for the rooted plane binary trees (interpretation d in Stanley's exercise 19) with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.

Original entry on oeis.org

0, 20, 2020, 2200, 202020, 202200, 220020, 220200, 222000, 20202020, 20202200, 20220020, 20220200, 20222000, 22002020, 22002200, 22020020, 22020200, 22022000, 22200020, 22200200, 22202000, 22220000, 2020202020, 2020202200
Offset: 0

Views

Author

Antti Karttunen, May 14 2002

Keywords

Links

Crossrefs

a(n) = 2*A063171(n) = A071153(A057123(n)).
Cf. A014486, A059984, A059985, A071153, A071154, A079436.

Programs

  • Mathematica
    balancedQ[0] = True; balancedQ[n_] := (s = 0; Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0]); 2*FromDigits /@ IntegerDigits[ Select[Range[0, 684], balancedQ], 2] (* Jean-François Alcover, Jul 24 2013 *)
Array[Map[FromDigits[# /. -1->0]*20 &, Select[Permutations[Join[Table[-1, #-1], Table[1,#]]], Min[Accumulate[#]] >=0 &]]&, 6, 0] (* Paolo Xausa, Mar 12 2024 *)
  • Python
    from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A071152_gen(): # generator of terms
    yield 0
    for l in count(1):
        for s in multiset_permutations('0'*l+'1'*(l-1)):
            c, m = 0, (l<<1)-1
            for i in range(m):
                if s[i] == '1':
                    c += 2
                if cA071152_list = list(islice(A071152_gen(),30)) # Chai Wah Wu, Nov 28 2023

Formula

a(n) = 2*A063171(n).

A209644 Łukasiewicz words (without the last zero) for rooted plane trees where non-leaf branching can occur only at the leftmost branch of any level, but nowhere else.

Original entry on oeis.org

0, 1, 20, 11, 300, 210, 120, 111, 4000, 3100, 2200, 1300, 2110, 1210, 1120, 1111, 50000, 41000, 32000, 23000, 14000, 31100, 22100, 13100, 21200, 12200, 11300, 21110, 12110, 11210, 11120, 11111, 600000, 510000, 420000, 330000, 240000, 150000, 411000, 321000, 231000, 141000, 312000, 222000, 132000, 213000
Offset: 0

Views

Author

Antti Karttunen, Mar 24 2012

Keywords

Comments

Note: this finite decimal representation works only up to the 511th term, as the 512th such word is already (10,0,0,0,0,0,0,0,0,0).

Links

Crossrefs

A209643 gives the positions of these terms in A071153 (A014486).
Cf. A071160.

Formula

a(n) = A071153(A209643(n)).
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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