A195697 First denominator and then numerator in a fraction expansion of log(2) - Pi/8.

2, 1, 3, -1, 12, 1, 30, 1, 35, -1, 56, 1, 90, 1, 99, -1, 132, 1, 182, 1, 195, -1, 240, 1, 306, 1, 323, -1, 380, 1, 462, 1, 483, -1, 552, 1, 650, 1, 675, -1, 756, 1, 870, 1, 899, -1, 992, 1, 1122, 1, 1155, -1, 1260

Offset

1,1

Comments

The minus sign in front of a fraction is considered the sign of the numerator.

References

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Links

Table of n, a(n) for n=1..53.

Formula

log(2) - Pi/8 = Sum_{n>=1} (-1)^(n+1)*(1/n) + (-1/2)*Sum_{n>=0} (-1)^n*(1/(2*n+1)).

Empirical g.f.: x*(2+x+x^2-2*x^3+9*x^4+2*x^5+14*x^6-2*x^7+3*x^8+2*x^9+3*x^10-2*x^11+x^13) / ((1-x)^3*(1+x)^3*(1-x+x^2)^2*(1+x+x^2)^2). - Colin Barker, Dec 17 2015

Example

1/2 - 1/3 + 1/12 + 1/30 - 1/35 + 1/56 + 1/90 - 1/99 + 1/132 + 1/182 - 1/195 + 1/240 + ... = [(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + (1/9 - 1/10) + (1/11 - 1/12) + ... ] - (1/2)*[(1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + (1/13 - 1/15) + ... ] = log(2) - Pi/8.

Crossrefs

Cf. A164833, A195908, A195909, A195913, A016655, A019675, A075549, A098289, A118324, A144981, A161685, A168056, A004772.

Sequence in context: A122048 A046208 A195909 * A137764 A344752 A057082

Adjacent sequences: A195694 A195695 A195696 * A195698 A195699 A195700

Keyword

frac,sign

Author

Mohammad K. Azarian, Sep 25 2011

Status

approved