A110976 -id:A110976 - cp's OEIS frontend

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

Giorgio Balzarotti and Paolo P. Lava, Oct 04 2005

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.

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

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

Cf. A051699, A110976, A110977.

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:

see program

A110977 Sequence of denominators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 9, 13, 9, 22, 9, 31, 71, 102, 71, 173, 71, 244, 559, 803, 559, 1362, 3283, 11211, 25705, 36916, 25705, 62621, 25705, 88326, 202357, 695397, 1593151, 2288548, 1593151, 3881699, 9356549, 13238248, 9356549, 22594797, 9356549

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Sep 28 2005

			if n = 2, B(n) = B(2) = 1 because B(0) = 1, B(1) = 1 * B(0) = 1 as the distances of n from closest prime are 2, 1, 0, 0, 1 ...

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

Cf. A051699, A109139, A109140, A110976.

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;

See program.

cp's OEIS Frontend

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).

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