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A000989
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3-adic valuation of binomial(2*n, n): largest k such that 3^k divides binomial(2*n, n).
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7
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0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3
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OFFSET
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0,6
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COMMENTS
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sign(a(n+1) - a(n)) is repeat [0, 1, -1]. - Filip Zaludek, Oct 29 2016
By Kummer's theorem, number of carries when adding n + n in base 3. - Robert Israel, Oct 30 2016
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LINKS
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FORMULA
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a(n) = Sum_{k>=0} floor(2*n/3^k) - 2*Sum_{k>=0} floor(n/3^k). - Benoit Cloitre, Aug 26 2003
If 2*n < 3^k then a(3^k+n) = a(n).
If n < 3^k < 2*n then a(3^k+n) = a(n)+1.
If n < 3^k then a(2*3^k+n) = a(n)+1. (End)
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MAPLE
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f:= proc(n) option remember; local k, m, d;
k:= floor(log[3](n));
d:= floor(n/3^k);
m:= n-d*3^k;
if d = 2 or 2*m > 3^k then procname(m)+1
else procname(m)
fi
end proc:
f(0):= 0:
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MATHEMATICA
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p=3; Array[ If[ Mod[ bi=Binomial[ 2#, # ], p ]==0, Select[ FactorInteger[ bi ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 27*3, 0 ]
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PROG
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(PARI) a(n) = valuation(binomial(2*n, n), 3)
(Haskell)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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