A186034 2-adic valuation of the n-th Motzkin number.

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

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..4095

Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.

Formula

a(n) = log_2(A001006(n)/numerator(A001006(n)/2^n)).

(-1)^a(n) = A186035(n).

a(n) = A007814(A001006(n)). - Antti Karttunen, Aug 12 2017

From Amiram Eldar, Aug 26 2024: (Start)

Results from Rowland and Yassawi (2015):

0 <= a(n) <= 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k=0..2} d(k) * k = 1/2, where d(k) is the asymptotic density of the occurrences of k in this sequence: d(0) = 2/3, and d(1) = d(2) = 1/6. (End)

Maple

A186034 := proc(n)
    local m, ml ;
    m := A001006(n) ;
    ml := %/2^n ;
    m/numer(ml) ;
    ilog2(%) ;
end proc:
seq(A186034(n), n=0..80) ; # R. J. Mathar, Feb 13 2015

Mathematica

IntegerExponent[RecurrenceTable[(n + 4) M[n + 2] - (2 n + 5) M[n + 1] - 3 (n + 1) M[n] == 0 && M[0] == M[1] == 1, M[n], {n, 0, 127}], 2] (* Eric Rowland, May 06 2013 *)

Prog

(Python)
from itertools import count, islice
def A186034_gen(): # generator of terms
    a, b = 1, 1
    yield from (0, 0)
    for n in count(2):
        a, b = b, (b*(2*n+1)+a*3*(n-1))//(n+2)
        yield (~b&b-1).bit_length()
A186034_list = list(islice(A186034_gen(), 30)) # Chai Wah Wu, Jul 08 2022

Crossrefs

Cf. A001006, A007814, A186035.

Sequence in context: A103270 A087780 A082523 * A280843 A221146 A083935

Adjacent sequences: A186031 A186032 A186033 * A186035 A186036 A186037

Keyword

nonn

Author

Paul Barry, Feb 11 2011

Extensions

Definition edited by Eric Rowland, May 06 2013

More terms from Antti Karttunen, Aug 12 2017

Status

approved