A072450 Continued fraction expansion of the limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

1, 1, 3, 7, 1, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 2, 20, 1, 2, 2, 2, 1, 2, 1, 1, 6, 2, 8, 5, 13, 2, 3, 1, 1, 115, 1, 4, 38, 4, 3, 1, 2, 1, 1, 1, 14, 1, 10, 4, 4, 5, 2, 2, 3, 19, 1, 1, 1, 5, 2, 1, 4, 1, 3, 1, 3, 4, 1, 8, 47, 33, 1, 1, 5, 13, 1, 14, 1, 5, 1, 1, 2, 17, 2, 1, 108, 9, 16, 3, 1, 2, 2, 3, 1, 5, 6, 2

Offset: 0

Author

Robert G. Wilson v, Aug 01 2002

Keywords

• cofr
• nonn

Comments

Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... = 1.75793275661800...

Increasing partial continued fractions of the above are 1, 3, 7, 20, 115, 233, 301, 328, 16902, ...

References

• Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
• David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 30.

Links

• Eric Weisstein's World of Mathematics, Nested Radical.

Crossrefs

Cf. A072449 (decimal expansion).

Programs

Mathematica

ContinuedFraction[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 100]

Extensions

Offset changed by Andrew Howroyd, Jul 06 2024