A156164 Decimal expansion of 17 + 12*sqrt(2).

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

Offset

2,1

Comments

Conjecturally, the fractional part 0.97056 27484 ... of this constant equals ( (1 + 2 * Sum_{n >= 1} (-1)^n*exp(-2*Pi*n^2))/(1 + 2 * Sum_{n >= 1} exp(-2*Pi*n^2)) )^4. The series are rapidly converging. For example, summing both series from n = 1 to n = 2 approximates the fractional part of the constant as ( (1 - 2*exp(-2*Pi) + 2*exp(-8*Pi))/(1 + 2*exp(-2*Pi) + 2*exp(-8*Pi)) )^4 = 0.97056 27484 77140 58562 026(89) ..., correct to 23 decimal places. - Peter Bala, Jun 05 2019

Links

G. C. Greubel, Table of n, a(n) for n = 2..10000

Maxim Kontsevich and Don Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in Mathematics Unlimited - 2001 and Beyond, B. Engquist and W. Schmid, editors, 2 vols., Springer-Verlag (2001), 771-808. See Section 2.4 (Example 4).

Index entries for algebraic numbers, degree 2.

Formula

Equals lim_{n -> oo} b(n)/b(n-1) for b = A156160, A156161, A156162, A278310.

Equals (3+2*sqrt(2))^2 = (1+sqrt(2))^4. - Klaus Brockhaus, Feb 14 2009. [corrected by Bruno Berselli, Feb 19 2013]

Minimal polynomial: x^2 - 34*x + 1. - Amiram Eldar, Apr 28 2026

Example

33.97056274847714058562026469051637694283606250452337687...

Mathematica

RealDigits[17 + 12 Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, Nov 30 2014 *)

Prog

(PARI) 17+sqrt(288) \\ Charles R Greathouse IV, May 07 2015

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17).

Cf. A005259, A059415/A059416, A156160, A156161, A156162, A278310.

Sequence in context: A165351 A215665 A200494 * A198613 A197031 A375069

Adjacent sequences: A156161 A156162 A156163 * A156165 A156166 A156167

Keyword

nonn,cons

Author

Klaus Brockhaus, Feb 09 2009

Status

approved