|
%I #29 Jul 09 2022 11:10:26
%S 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,
%T 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,
%U 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
%N 2-adic valuation of the n-th Motzkin number.
%H Antti Karttunen, <a href="/A186034/b186034.txt">Table of n, a(n) for n = 0..4095</a>
%H E. Rowland and R. Yassawi, <a href="http://arxiv.org/abs/1310.8635">Automatic congruences for diagonals of rational functions</a>, arXiv preprint arXiv:1310.8635 [math.NT], 2013-2014.
%F a(n) = log_2(A001006(n)/numerator(A001006(n)/2^n)).
%F (-1)^a(n) = A186035(n).
%F a(n) = A007814(A001006(n)). - _Antti Karttunen_, Aug 12 2017
%p A186034 := proc(n)
%p local m,ml ;
%p m := A001006(n) ;
%p ml := %/2^n ;
%p m/numer(ml) ;
%p ilog2(%) ;
%p end proc:
%p seq(A186034(n),n=0..80) ; # _R. J. Mathar_, Feb 13 2015
%t 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 *)
%o (Python)
%o from itertools import count, islice
%o def A186034_gen(): # generator of terms
%o a, b = 1, 1
%o yield from (0, 0)
%o for n in count(2):
%o a, b = b, (b*(2*n+1)+a*3*(n-1))//(n+2)
%o yield (~b&b-1).bit_length()
%o A186034_list = list(islice(A186034_gen(),30)) # _Chai Wah Wu_, Jul 08 2022
%Y Cf. A001006, A007814, A186035.
%K nonn
%O 0,4
%A _Paul Barry_, Feb 11 2011
%E Definition edited by _Eric Rowland_, May 06 2013
%E More terms from _Antti Karttunen_, Aug 12 2017
|
