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Search: id:A166711
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| A166711 |
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Permutation of the integers: two positives, one negative. |
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+0
4
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| 0, 1, 2, -1, 3, 4, -2, 5, 6, -3, 7, 8, -4, 9, 10, -5, 11, 12, -6, 13, 14, -7, 15, 16, -8, 17, 18, -9, 19, 20, -10, 21, 22, -11, 23, 24, -12, 25, 26, -13, 27, 28, -14, 29, 30, -15, 31, 32, -16, 33, 34, -17, 35, 36, -18, 37, 38, -19, 39, 40, -20, 41, 42, -21, 43, 44, -22, 45, 46
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Setting m=2 in
log(m) = Sum_{n>0} (n mod m - (n-1) mod m)/n [1]
yields the sum
log(2) = (1 -1/2) +(1/3 -1/4) +(1/5 -1/6)+...
Substituting every -1/d by 1/d - 2/d we obtain
log(2) = (1+1/2-1)+(1/3+1/4-1/2)+(1/5+1/6-1/3)+...
a(n) is the sequence of denominators of this modified sum with unit numerators, so
Sum_{k>0} 1/a(k) = log(2)
Substituting -1/d by -2/d + 1/d would yield another permutation (one positive, one negative, one positive) with the same sum of inverses.
Similar sequences (m positives, one negative) may be obtained for the logarithm of any integer m>0. A001057 is the case m=1, with sum of inverses log(1).
Equation [1] is a result of expanding log( Sum_{0<=k<=m-1} x^k ) at x=1 (See comment to A061347.)
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LINKS
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Wikipedia, Riemann series theorem [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 01 2009]
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FORMULA
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G.f.: (x*(1+2*x-x^2+x^3)/((1-x)^2*(1+x+x^2)^2))
a(0)=0, a(1)=1, a(2)=2, a(3)=-1, a(4)=3, a(5)=4, a(n)=2*a(n-3)-a(n-6), n>=6.
a(n) = (n+1)/3 +2*A049347(n)/3 -(-1)^n*A076118(n+1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009]
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PROGRAM
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(PARI) a(n)=(2*(n+1)\3)*(1-3/2*!(n%3))
(PARI) a(n)=if(n>=0, [ -n\3, 2*(n\3)+1, 2*(n\3)+2][n%3+1]) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 14 2009]
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CROSSREFS
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Cf. A001057, A002162, A038608. Signed and shifted version of A009947.
Sequence in context: A117384 A125160 A009947 this_sequence A026249 A130527 A026366
Adjacent sequences: A166708 A166709 A166710 this_sequence A166712 A166713 A166714
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KEYWORD
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sign,new
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AUTHOR
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Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 18 2009
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EXTENSIONS
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Corrected by Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 22 2009
frac keyword removed Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 02 2009
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