A119811 Numerators of the convergents to the continued fraction for the constant A119809 defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

2, 7, 72, 9511, 1246930216, 2742028548141904733479, 1737967067447512977484869808775151193351704374584616

Offset: 1

Author

Paul D. Hanna, May 26 2006

Keywords

• frac
• nonn

Comments

The number of digits in these numerators are (beginning at n=1): [1,1,2,4,10,22,52,124,297,717,1729,4173,10074,24319,58709,141735,..].

Examples

			c = 2.32258852258806773012144068278798408011950250800432925665718...
Convergents begin:
[2/1, 7/3, 72/31, 9511/4095, 1246930216/536870911,...]
where the denominators of the convergents equal [2^A000129(n-1)-1]:
[1,3,31,4095,536870911,1180591620717411303423,...],
and A000129 is the Pell numbers.
		

Crossrefs

Cf. A119809 (constant), A119811 (continued fraction), A000129; A119812 (dual constant).

Programs

PARI

{a(n)=local(M=contfracpnqn(vector(n,k,if(k==1,2, 2^round(((1+sqrt(2))^(k-1)+(1-sqrt(2))^(k-1))/2) +2^round(((1+sqrt(2))^(k-2)-(1-sqrt(2))^(k-2))/(2*sqrt(2))))))); return(M[1,1])}