A099879 Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...

1, 9, 4, 2, 6, 5, 5, 4, 2, 2, 7, 6, 3, 9, 8, 7, 3, 2, 8, 2, 2, 1, 4, 1, 3, 2, 9, 1, 4, 1, 2, 6, 6, 7, 2, 3, 7, 6, 8, 8, 0, 7, 3, 6, 3, 0, 0, 0, 7, 1, 1, 5, 5, 1, 5, 1, 0, 0, 5, 6, 9, 5, 6, 1, 7, 7, 7, 6, 3, 2, 2, 3, 0, 8, 8, 9, 3, 4, 3, 4, 6, 5, 9, 7, 1, 5, 3, 2, 0, 8, 0, 6, 7, 5, 0, 1, 6, 8, 5, 3, 9, 8, 0

Offset: 1

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 03 2004

Keywords

• cons
• easy
• nonn

Examples

			1.94265542276398732822141329141266723768807363...
		

Links

• Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Crossrefs

Cf. A072449, A099874, A099876 to A099878 for other nested radicals.

Programs

Mathematica

k = 64; r = 65; While[k > 0, r = Sqrt[k^2 + r]; k-- ]; RealDigits[r, 10, 111][[1]] (* Robert G. Wilson v, Nov 04 2004 *)

PARI

t=0; forstep(n=100,1,-1,t=sqrt(t+n^2)); print(t)

PARI

\\ We need about b/log(b) steps, where epsilon = 2^-b.
my(b=bitprecision(1.),t); forstep(n=b\log(b)+9,1,-1, t=sqrt(t+n^2)); t \\ Charles R Greathouse IV, Aug 19 2025