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%I #52 Dec 27 2023 11:53:30
%S 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,
%T 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,
%U 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
%N Decimal expansion of 1/7.
%C 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
%C 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
%C Decimal expansion of -99*(zeta(-5) + zeta(-9)) - 1. - _Arkadiusz Wesolowski_, Sep 15 2013
%C Also, decimal expansion of Sum_{i>0} 1/8^i. - _Bruno Berselli_, Jan 03 2014
%C 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
%D H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrüche'.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.
%H J. Hall, <a href="http://mathworld.wolfram.com/One-SeventhEllipse.html">One-Seventh Ellipse,</a> on MathWorld, by E. W. Weisstein.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1).
%F A028416(1)=7; A002371(A049084(7)) = A002371(4) = 6. a(n+6) = a(n), a(n+6/2) = 9 - a(n). - _Reinhard Zumkeller_, Oct 06 2008
%F From _Colin Barker_, Aug 14 2012: (Start)
%F a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
%F G.f.: (1+3*x-2*x^2+7*x^3) / ((1-x)*(1+x)*(1-x+x^2)). (End)
%F a(n) = A068028(n+2). - _Zak Seidov_, Mar 26 2015
%F 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 0.142857142857142857...
%p Digits:=100: evalf(1/7); # _Wesley Ivan Hurt_, Jun 28 2016
%t 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 *)
%o (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
%o (PARI) 1/7. \\ _Charles R Greathouse IV_, Sep 24 2015
%o (PARI) digits(10^99\7) \\ _M. F. Hasler_, Oct 25 2017
%Y Cf. A002371, A028416, A049084, A068028.
%K nonn,cons,easy
%O 0,2
%A _N. J. A. Sloane_
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