A110917 Conversion to a regular-simple continued-fraction approximation of the limit value (C0=2.7745963816360040537087...) of the continued fraction (numerator = A110976 and denominator = A110977) based on the sequence of the distances of n from closest primes (A051699).

2, 1, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 4, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 5, 4, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 4, 5

Offset: 1

Author

Giorgio Balzarotti and Paolo P. Lava, Oct 04 2005

Keywords

• cofr
• nonn

Comments

With the exception of n = 3, it should be abs(a(n)-a(n-1)) = < 1 for all n. Hill-mountain-like plot, with land = 2.

Examples

			C0 = a(1) +1/( a(2) +1/( a(3) +1/( a(4) +1/( a(5) +...=2+1/(1+1/(3+1/(2+1/(3+...
		

References

• G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 110.

Crossrefs

Cf. A051699, A110976, A110977.

Programs

Maple

cd:=proc(N) # d[n]distance of n from closest prime A[0]:=d[0]; A[1]:=d[1]*A[0]+1; B[0]:=1; B[1]:=d[1]*B[0]; for n from 2 by 1 to N do A[n]:=d[n]*A[n-1]+A[n-2]; B[n]:=d[n]*B[n-1]+B[n-2]; od; R:=A[N]/B[N]; convert(R,confrac); end:

Formula

see program