A166871 Permutation of the integers: 3 positives, 2 negatives.

0, 1, 2, 3, -1, -2, 4, 5, 6, -3, -4, 7, 8, 9, -5, -6, 10, 11, 12, -7, -8, 13, 14, 15, -9, -10, 16, 17, 18, -11, -12, 19, 20, 21, -13, -14, 22, 23, 24, -15, -16, 25, 26, 27, -17, -18, 28, 29, 30, -19, -20, 31, 32, 33, -21, -22, 34, 35, 36, -23, -24, 37, 38, 39, -25, -26, 40, 41

Offset: 0

Author

Jaume Oliver Lafont, Oct 22 2009

Keywords

• sign
• easy

Comments

This sequence enumerates the denominators with sign in case p=3 and n=2 of:

log(p/n) = sum( i>=0, sum(p*i+1<=j<=p*(i+1),1/j) - sum(n*i+1<=j<=n*(i+1),1/j) )

Similar sequences can be constructed for the logarithm of any rational r=p/n (p,n>0), enumerating p positive integers and n negative integers every p+n terms.

Case p=2, n=1 is A166711.

Case p=1, n=1 is A001057.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wikipedia, Riemann series theorem
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1)

Crossrefs

Cf. A016578, A166711.

Programs

Mathematica

LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,1,2,3,-1,-2,4,5,6,-3}, 100] (* G. C. Greubel, May 27 2016 *)

Formula

Sum_{k>0} 1/a(k) = log(3/2).

G.f.: x*(1+2*x+3*x^2-x^3-2*x^4+2*x^5+x^6-x^8)/((x-1)^2*(x^4+x^3+x^2+x+1)^2 ).

Extensions

keyword frac removed Jaume Oliver Lafont, Nov 02 2009