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A021085 Decimal expansion of 1/81. 11

%I #37 Jun 04 2021 22:08:13

%S 0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,

%T 7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,

%U 5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9,0,1,2,3,4,5,6,7,9

%N Decimal expansion of 1/81.

%C The decimal expansion of Sum_{n>=1} floor(n * tanh(Pi))/10^n is the same as that of 1/81 for the first 268 decimal places [Borwein et al.]

%C Sqrt(999999999999999999) = 9*sqrt(12345679012345679). - _Ryohei Miyadera_, Ken Hirotomi, Hiroyuki Ozaki and Atushi Tanaka, Jan 16 2006

%D J. Borwein, D. Bailey and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Peters, Boston, 2004. See Sect. 1.4.

%H Jean-François Alcover, <a href="/A021085/a021085.txt">300 digits of Sum_{n>=1} floor(n*tanh(Pi))/10^n</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,1).

%F Equals Sum_{k >= 1} (1/2^k)*(1/5^k)*k. - _Eric Desbiaux_, Mar 11 2009

%F G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 9*x^7)/(1 - x^9). - _Ilya Gutkovskiy_, Jun 21 2016

%F From _Stefano Spezia_, Jun 03 2021: (Start)

%F a(n) = a(n-9) for n > 8.

%F Equals (1/10)*Sum_{n>0} 1/A052268(n). (End)

%t Table[Mod[n, 9], {n, 0, 120}] /. 8 -> 9 (* or *)

%t PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits[N[1/81, 120]] (* _Michael De Vlieger_, Jun 21 2016 *)

%t PadRight[{},120,{0,1,2,3,4,5,6,7,9}](* _Harvey P. Dale_, Apr 07 2019 *)

%Y Cf. A052268.

%K nonn,easy,cons

%O 0,3

%A _N. J. A. Sloane_

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