A154925 The terms of this sequence are integer values of consecutive denominators (with signs) from the fractional expansion (using only fractions with numerators to be positive 1's) of the BBP polynomial ( 4/(8*k+1) - 2/(8*k+4) - 1/(8*k+5) - 1/(8*k+6) ) for all k (starting from 0 to infinity).

1, 1, 1, 1, -2, -5, -6, 3, 9, -5, -13, -14, 5, 30, 510, -10, -21, -22, 7, 59, 5163, 53307975, -14, -29, -30

Offset: 0

Author

Alexander R. Povolotsky, Jan 17 2009, Jan 18 2009

Keywords

• sign
• uned

Comments

The Egyptian fraction expansion is applied to the first fraction (that is, 4/(8*k+1) ) of the BBP polynomial ( 4/(8*k+1) - 2/(8*k+4) - 1/(8*k+5) - 1/(8*k+6) ) for k >= 1. R. Knott's converter calculator #1 (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#calc1) is used for such conversion. Note that in the case of k=0, 4/(8*k+1) = 4 and could be trivially expressed as 1/1 + 1/1 + 1/1 + 1/1. It remains to be seen how the above described Pi presentation relates to Engel's presentation of Pi, which also consists of an infinite sum of fractions whose numerators are all 1's.

Examples

			For k=1, 4/(8*k+1) = 4/9 = 1/3 + 1/9, thus the first (smallest) denominator is 3 so a(7)=3.
For k=1, 4/(8*k+1) = 4/9 = 1/3 + 1/9 and the second (next to smallest) denominator is 9 so a(8)=9.
		

Links

• Wikipedia, Bailey-Borwein-Plouffe formula.

Crossrefs

Cf. A154429.