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A186034
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2-adic valuation of the n-th Motzkin number.
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2
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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
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OFFSET
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0,4
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..4095
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013-2014.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001006, A007814, A186035.
Sequence in context: A103270 A087780 A082523 * A280843 A221146 A083935
Adjacent sequences: A186031 A186032 A186033 * A186035 A186036 A186037
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KEYWORD
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nonn
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AUTHOR
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Paul Barry, Feb 11 2011
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EXTENSIONS
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Definition edited by Eric Rowland, May 06 2013
More terms from Antti Karttunen, Aug 12 2017
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STATUS
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approved
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