A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

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

Offset: 1

Author

Robert G. Wilson v, Oct 03 2018

Keywords

• nonn
• cons
• easy

Comments

For x >= 0, sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)))) = (sqrt(4*x+1) + 1)/2. This is an integer for each x such that 2*x is a term in A000217.

Examples

			3.541381265149109844499842122601033531042485047393205593209576523243166362659...
		

Links

• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A000007, A001622, A000038, A209927, A222132, A222134, A223140, A235162.

Programs

Maple

evalf((sqrt(37)+1)/2,120); # Muniru A Asiru, Oct 07 2018

Mathematica

RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[9, {135}]], 10, 111][[1]] (* or *)
RealDigits[(Sqrt[37] + 1)/2, 10, 111][[1]]

PARI

(sqrt(37)+1)/2 \\ Altug Alkan, Oct 03 2018

Formula

Minimal polynomial: x^2 - x - 9. - Stefano Spezia, Jul 02 2025