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Search: id:A004128
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| A004128 |
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Sum_{k=1..n} floor(3n/3^k). |
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+0
15
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| 0, 1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 30, 31, 32, 34, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 72, 74, 75, 76, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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3-adic valuation of (3n)! - cf. A054861.
Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are in |A067622|.
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REFERENCES
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Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth and Number", by Jay Kappraff, World Scientific, 2002, p. 356.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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a(n) = n+[n/3]+[n/9]+[n/27]+... = n+a([n/3]) = n+A054861(n) = A054861(3n) = (3n-A053735(n))/2. - Henry Bottomley (se16(AT)btinternet.com), May 01 2001
a(n)=sum{k>=0, floor(n/3^k)}. a(n)=sum{0<=k<=floor(log_3(n)), floor(n/3^k)}, n>=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Recurrence: a(n)=n+a(floor(n/3)); a(3n)=3n+a(n); a(n*3^m)=3*n*(3^m-1)/2+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(k*3^m)=k*(3^(m+1)-1)/2, 0<=k<3, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
Asymptotic behavior: a(n)=3/2*n+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)<=(3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
a(n)>=(3n-2)/2-floor(log_3(n)); equality holds for n=3^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim inf (3n/2-a(n))=1/2, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (3n/2-log_3(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
lim sup (a(n+1)-a(n)-log_3(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
G.f.: g(x)=sum{k>=0, x^(3^k)/(1-x^(3^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
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PROGRAM
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(PARI) a(n)=local(s, t=1); while(t<=n, s+=n\t; t*=3); s - Michael Somos Feb 26 2004
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CROSSREFS
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Cf. A004117, A001511, A051064, A055457.
A051064(n) = a(n+1) - a(n). - Alford Arnold (Alford1940(AT)aol.com), Jul 19 2000
Cf. A054861, A067080, A098844, A132027, A005187, A054899.
Sequence in context: A091529 A095775 A035063 this_sequence A023717 A043687 A087118
Adjacent sequences: A004125 A004126 A004127 this_sequence A004129 A004130 A004131
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Current definition suggested by Jason Earls (zevi_35711(AT)yahoo.com), Jul 04 2001
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