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

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

Offset: 1

Author

Amiram Eldar, Aug 06 2021

Keywords

• nonn
• cofr
• base

Comments

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

Examples

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

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.

Crossrefs

Cf. A269707.

Programs

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]

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.