A188884 Continued fraction of (1 + sqrt(1 + Pi^2))/Pi.

1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 7, 1, 1, 23, 2, 2, 4, 3, 1, 11, 158, 1, 1, 1, 1, 4, 2, 1, 6, 2, 19, 75, 1, 1, 1, 28, 1, 29, 6, 8, 1, 5, 1, 4, 2, 1, 8, 1, 1, 19, 1, 1, 9, 2, 2, 3, 1, 2, 11, 1, 1, 3, 1, 1, 4, 169, 1, 1, 2, 1, 3, 1, 1, 10, 2, 1, 3, 8, 2, 4, 8, 5, 1, 8, 1, 7, 1, 1, 1, 1, 4, 38, 1, 5, 1, 43, 1, 1, 1, 1, 2, 1, 8, 1, 20, 1, 1, 1, 2, 13, 51, 2, 21, 1, 2, 5, 1, 1, 1

Offset: 0

Author

Clark Kimberling, Apr 12 2011

Keywords

• nonn
• cofr

Comments

For a geometric interpretation, see A188640 and A188883.

Examples

			(1 + sqrt(1 + Pi^2))/Pi = [1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 7, 1, 1, 23, 2, ...].
		

Crossrefs

Cf. A188640, A188883 (decimal expansion), A188726.

Programs

Mathematica

r = 2/Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[(1+Sqrt[1+Pi^2])/Pi,120] (* Harvey P. Dale, May 02 2025 *)

Extensions

Offset changed by Andrew Howroyd, Jul 07 2024