A228788 - OEIS

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A228788

Decimal expansion of the algebraic integer 2*cos(Pi/34) of degree 16 = A055034(34) (over the rationals), the length ratio (smallest diagonal)/side of a regular 34-gon.

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

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OFFSET

1,2

COMMENTS

rho(34):= 2*cos(Pi/34) is used in the algebraic number field Q(rho(34)) of degree 16 (see A187360) in which s(17) = 2*cos(Pi/17) (for its decimal expansion see A228787), the length ratio side/R of a regular 17-gon inscribed in a circle of radius R, is an integer. See A228787 for this expansion.

Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into rho(34) = sqrt(2+sqrt(2+2*cos(2*Pi/17))).

The minimal polynomial of rho(34) is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16 (row n=34 polynomial of A187360).

The continued fraction expansion starts with 1; 1, 116, 4, 1, 2, 1, 20, 2, 2, 1, 7, 10, 2, 2, 1, 3, 6, 1, 4, 4, 15, ...

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Kival Ngaokrajang, Illustration of the length ratio (smallest diagonal)/side of a regular 34-gon

FORMULA

2*cos(Pi/34) = 1.99146835259006904374238235781096...

MATHEMATICA

RealDigits[2 Cos[Pi/34], 10, 111][[1]] (* Robert G. Wilson v, Jul 28 2014 *)

PROG

(PARI) 2*cos(Pi/34) \\ Charles R Greathouse IV, Nov 12 2014

CROSSREFS

Cf. A055034, A187360, A210644, A228787.