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A208060
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a(n) = 1 + 2*n + 2^2*n*[n/2] + 2^3*n*[n/2]*[n/3] + 2^4*n*[n/2]*[n/3]*[n/4] + ... where [x]=floor(x).
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3
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1, 3, 13, 43, 233, 611, 4405, 10515, 64145, 218755, 1215821, 2689083, 28162105, 61179795, 307475813, 1236997051, 8042542625, 17101581699, 146671231501, 309740445795, 2415132010441, 8877053064643, 40919003272005, 85564885298027, 1068638260341937, 2783025471994851
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OFFSET
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0,2
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COMMENTS
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Compare the definition of a(n) to the exponential series:
exp(2*n) = 1 + 2*n + 2^2*n*(n/2) + 2^3*n*(n/2)*(n/3) + 2^4*n*(n/2)*(n/3)*(n/4) + ...
Conjecture: limit a(n)^(1/n) = 2*L where L = 2.200161058099... is the geometric mean of Luroth expansions, where log(L) = Sum_{n>=1} log(n)/(n*(n+1)) = 0.7885305659115... (cf. A085361).
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} 2^k*floor(n/k).
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EXAMPLE
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a(5) = 1 + 2*5+ 4*5[5/2] + 8*5[5/2][5/3] + 16*5[5/2][5/3][5/4] + 32*5[5/2][5/3][5/4][5/5] = 1 + 2*5 + 4*5*2 + 8*5*2*1 + 16*5*2*1*1 + 32*5*2*1*1*1 = 611.
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PROG
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(PARI) {a(n)=1+sum(m=1, n, prod(k=1, m, 2*floor(n/k)))}
(PARI) /* More efficient: variant of a program by Charles R Greathouse IV */
{a(n)=my(k=1); 1+sum(m=1, n, k*=2*(n\m))}
for(n=0, 60, print1(a(n), ", "))
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CROSSREFS
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Cf. A075885.
Sequence in context: A019008 A062609 A004660 * A284194 A209296 A193041
Adjacent sequences: A208057 A208058 A208059 * A208061 A208062 A208063
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Feb 23 2012
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STATUS
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approved
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