A346909 Continued fraction expansion of the constant whose decimal expansion is A269707.

3, 3, 3, 30, 330, 303000, 33003300000, 3030000030300000000000, 3300330000000000330033000000000000000000000, 30300000303000000000000000000000303000003030000000000000000000000000000000000000000000

Offset

1,1

Comments

The next term has 171 digits and is too large to include in the Data section.

References

André Blanchard and Michel Mendès France, Symétrie et transcendance, Bull. Sc. Math., 2nd series, Vol. 106 (1982), pp. 325-335.

Links

Amiram Eldar, Table of n, a(n) for n = 1..13

M. Mendes France and A. J. van der Poorten, Some explicit continued fraction expansions, Mathematika, Vol. 38, No. 1 (1991), pp. 1-9.

Formula

a(n) = 3 * 10^((4^((n-3)/2)-1)/3) * Product_{k=0..(n-5)/2} (1 + 10^(4^k)), if n > 2 is odd, and 3 * 10^((2*4^(n/2-2)+1)/3) * Product_{k=0..n/2-3} (1 + 10^(2*4^k)), if n > 2 is even.

Example

3 + 1/(3 + 1/(3 + 1/(30 + 1/(330 + ... )))) = 3.30033000000000033... (A269707).

Mathematica

a[1] = a[2] = 3; a[n_] := 3 * If[OddQ[n], 10^((4^((n - 3)/2) - 1)/3) * Product[1 + 10^(4^k), {k, 0, (n - 5)/2}], 10^((2*4^(n/2 - 2) + 1)/3) * Product[1 + 10^(2*4^k), {k, 0, n/2 - 3}]]; Array[a, 10]

Crossrefs

Cf. A269707.

Sequence in context: A230495 A372019 A369081 * A369232 A369014 A025549

Adjacent sequences: A346906 A346907 A346908 * A346910 A346911 A346912

Keyword

nonn,cofr,base

Author

Amiram Eldar, Aug 06 2021

Status

approved