A340131 -id:A340131 - cp's OEIS frontend

A340395 a(n) = A340131(A001006(n)).

5, 15, 50, 150, 455, 1365, 4100, 12300, 36905, 110715, 332150, 996450, 2989355, 8968065, 26904200, 80712600, 242137805, 726413415, 2179240250, 6537720750, 19613162255, 58839486765, 176518460300, 529555380900, 1588666142705, 4765998428115, 14297995284350

Offset: 2

Gennady Eremin, Jan 06 2021

This sequence is associated with A340131, whose terms are sorted by the length of their ternary code. Elements with the same length of ternary code form a range that has a maximum. The maximal term of the n-range (a set of elements with ternary code length n in A340131) is a(n). Example: numbers 29, 33, 44, 45 and 50 have a ternary length of 4 (see A340131), respectively a(4) = 50.

Ternary code for a(n) is 12..12 for even n and 12..120 for odd n.

			A001006(2) = 2, so a(2) = A340131(2) = 5.
A001006(3) = 4, so a(3) = A340131(4) = 15, etc.

Andrew Howroyd, Table of n, a(n) for n = 2..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Subsequence of A340131.

Vec(5/(1 - 3*x - x^2 + 3*x^3) + O(x^30)) \\ Andrew Howroyd, Jan 08 2021

a(n) = 5*3^(n-2*k)*(9^k-1)/8 where k = floor(n/2).
a(n+1) = 3*a(n) for even n >= 2; a(n+1) = 3*a(n)+5 for odd n >= 3.
a(n) = 5*A033113(n-1).
a(n) = (5/8)*(3^n - (-1)^(n-1) - 2).
a(n) = 2*a(n-1) + 3*a(n-2) + 5 for n > 3.
From Stefano Spezia, Jan 06 2021: (Start)
G.f.: 5*x^2/(1 - 3*x - x^2 + 3*x^3).
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 4. (End)

A340544 Numbers from A340131 that are not multiples of 3.

Original entry on oeis.org

5, 11, 29, 44, 50, 83, 98, 104, 116, 128, 140, 146, 245, 260, 266, 278, 290, 302, 308, 332, 344, 377, 380, 395, 401, 410, 416, 434, 449, 455, 731, 746, 752, 764, 776, 788, 794, 818, 830, 863, 866, 881, 887, 896, 902, 920, 935, 941, 980, 992, 1025, 1028, 1043

Offset: 1

Gennady Eremin, Jan 11 2021

Terms are reduced, i.e., ternary codes do not have trailing zeros.
The term is a digitized Motzkin path that starts with an up step and ends with a down step. Such a path has neither leading nor final flat steps, i.e., the ternary code of the corresponding term has no finite 0's. Recall that in ternary code, 1's are up steps, and 2's are down steps.
The number of terms with a ternary code of length k is A026107(k-1). For instance, 7 (seven) reduced terms 83, 98, 104, 116, 128, 140, and 146 have a ternary length of 5, namely 10002, 10122, 10212, 11022, 11202, 12012, and 12102. Respectively A026107(4) = 7.

Gennady Eremin, Table of n, a(n) for n = 1..1000
Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 105.

Intersection of A001651 and A340131.
Subsequences: A134752, A168607.
Cf. A026107.

def digits(n, b):
  out = []
  while n >= b:
    out.append(n % b)
    n //= b
  return [n] + out[::-1]
def ok(n):
  if n%3 == 0: return False
  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)]) # after Michael S. Branicky (A340131)

cp's OEIS Frontend

A340395 a(n) = A340131(A001006(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula