A020806 Decimal expansion of 1/7.

1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

142857 and 999999 = 7*142857 are first and last Kaprekar numbers with six digits. Note a(n) + a(n+3) = 9. (142857^2 = 20408122449; 20408 + 122449 = 142857.) a(n)^2 = 1, 16, 4, 64, 25, 49, ... - Paul Curtz, Aug 24 2009

The constant 19 + 1/7 = 19.142857... is the Kirchhoff index of the Möbius ladder graph on v=8 vertices. The Laplacian matrix has the eigenvalues 4 (one time), 4-sqrt(2) (2 times), 4+sqrt(2) (2 times), 2 (2 times) and 0 (one time). Then the Kirchhoff index is v times the sum over the inverse, nonzero eigenvalues. - R. J. Mathar, Feb 13 2011

Decimal expansion of -99*(zeta(-5) + zeta(-9)) - 1. - Arkadiusz Wesolowski, Sep 15 2013

Also, decimal expansion of Sum_{i>0} 1/8^i. - Bruno Berselli, Jan 03 2014

The points whose coordinates are overlapping pairs of digits of this sequence, (1, 4), (4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on one ellipse, with equation 19*x^2 + 36*x*y + 41*y^2 - 333*x - 531*y = -1638. Overlapping pairs of pairs of digits, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42), also yield 6 points on one ellipse, with equation -165104*x^2 + 160804*x*y + 8385498*x - 41651*y^2 - 3836349*y = 7999600. (See book by Wells and MathWorld link.) - M. F. Hasler, Oct 25 2017

Examples

			0.142857142857142857...
		

References

• H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrüche'.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

Links

• J. Hall, One-Seventh Ellipse, on MathWorld, by E. W. Weisstein.
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Cf. A002371, A028416, A049084, A068028.

Programs

Magma

I:=[1,4,2,8]; [n le 4 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

Maple

Digits:=100: evalf(1/7); # Wesley Ivan Hurt, Jun 28 2016

Mathematica

CoefficientList[Series[(1 + 3 x - 2 x^2 + 7 x^3) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
realDigitsRecip[7] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 18 2024 *)

PARI

1/7. \\ Charles R Greathouse IV, Sep 24 2015

PARI

digits(10^99\7) \\ M. F. Hasler, Oct 25 2017

Formula

From Reinhard Zumkeller, Oct 06 2008: (Start)

A028416(1)=7; A002371(A049084(7)) = A002371(4) = 6.

a(n+6) = a(n), a(n+6/2) = 9 - a(n). (End)

From Colin Barker, Aug 14 2012: (Start)

a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.

G.f.: (1+3*x-2*x^2+7*x^3) / ((1-x)*(1+x)*(1-x+x^2)). (End)

a(n) = A068028(n+2). - Zak Seidov, Mar 26 2015

a(n) = (27 - 11*cos(n*Pi) - 10*cos(n*Pi/3) - 6*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 28 2016

E.g.f.: (8*cosh(x) - exp(x/2)*(5*cos(sqrt(3)*x/2) + 3*sqrt(3)*sin(sqrt(3)*x/2)) + 19*sinh(x))/3. - Stefano Spezia, Dec 07 2024