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A340131 Numbers whose ternary expansions have the same number of 1's and 2's and, in each prefix (initial fragment), at least as many 1's as 2's. 3
0, 5, 11, 15, 29, 33, 44, 45, 50, 83, 87, 98, 99, 104, 116, 128, 132, 135, 140, 146, 150, 245, 249, 260, 261, 266, 278, 290, 294, 297, 302, 308, 312, 332, 344, 348, 377, 380, 384, 395, 396, 401, 405, 410, 416, 420, 434, 438, 449, 450, 455, 731, 735, 746, 747 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For a nonzero term, the ternary code starts with 1, otherwise the balance of 1's and 2's is broken already in the one-digit prefix. Therefore 7, 19, 21, etc. (see A039001) are not terms.

As another example, for the integer 52 the balance is broken in the three-digit prefix 122 (the entire ternary code is 1221).

Each term with a ternary code of length k corresponds one-to-one to the Motzkin path of length k that starts with an up step. Therefore, the terms can be called digitized Motzkin paths.

The number of terms with a ternary code of length k is equal to A244884(k). Example: five terms 29, 33, 44, 45 and 50 have a ternary length of 4, respectively A244884(4)=5.

LINKS

Gennady Eremin, Table of n, a(n) for n = 1..1000

Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.

EXAMPLE

The first terms 0 and 5 are obvious, because the four intermediate ternary codes 1, 2, 10[3], and 11[4] are rejected due to a violation of the balance of 1's and 2's. Next, the successor function S works: for any term x, the next term is S(x).

Iterating over numbers is inefficient; code suffixes (final digits) can be processed faster. The transition from 0 to 12[5] is generalized for terms that are multiples of 9. For example,

S(10200[99]) = 10212[104], S(1122000[1188]) = 1122012[1193], etc.

In this case, the calculation of the subsequent term is reduced to simply replacing the suffix s = 00 with the subsequent suffix s'= 12.

Another common suffix is s = 02..2 = 02^k (twos are repeated at the end of the ternary code). Then the subsequent suffix is s'= 202..2 = 202^(k-1), i.e., within such a suffix, the first two digits are reversed. Here are some examples:

k = 1, S(1002[29]) = 1020[33], the increment is 4*3^0 = 4;

k = 2, S(110022[332]) = 110202[344], the increment is 4*3^1 = 12;

k = 3, S(10110222[2537]) = 10112022[2573], the increment is 4*3^2 = 36;

k = 4, S(111102222[9800]) = 111120222[9908], the increment is 4*3^3 = 108.

There are 5 such group suffixes.

PROG

(PARI) is(n) = {my(d = digits(n, 3), v = [0, 0]); for(i = 1, #d, if(d[i] > 0, v[d[i]]++); if(v[1] < v[2], return(0))); v[1] == v[2] } \\ David A. Corneth, Dec 29 2020

(Python)

def digits(n, b):

  out = []

  while n >= b:

    out.append(n % b)

    n //= b

  return [n] + out[::-1]

def ok(n):

  t = digits(n, 3)

  if t.count(1) != t.count(2): return False

  return all(t[:i].count(1) >= t[:i].count(2) for i in range(1, len(t)))

print([n for n in range(750) if ok(n)]) # Michael S. Branicky, Dec 29 2020

CROSSREFS

Subsequence of A039001.

Subsequences: A134752, A168607.

Sequence in context: A314078 A314079 A137009 * A290199 A034905 A031153

Adjacent sequences:  A340128 A340129 A340130 * A340132 A340133 A340134

KEYWORD

nonn,easy,base

AUTHOR

Gennady Eremin, Dec 29 2020

STATUS

approved

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Last modified September 19 11:27 EDT 2021. Contains 347556 sequences. (Running on oeis4.)