A021085 Decimal expansion of 1/81.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• cons

Comments

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.]

Sqrt(999999999999999999) = 9*sqrt(12345679012345679). - Ryohei Miyadera, Ken Hirotomi, Hiroyuki Ozaki and Atushi Tanaka, Jan 16 2006

References

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

Links

• Jean-François Alcover, 300 digits of Sum_{n>=1} floor(n*tanh(Pi))/10^n
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A052268.

Programs

Mathematica

Table[Mod[n, 9], {n, 0, 120}] /. 8 -> 9 (* or *)
PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits[N[1/81, 120]] (* Michael De Vlieger, Jun 21 2016 *)
PadRight[{},120,{0,1,2,3,4,5,6,7,9}](* Harvey P. Dale, Apr 07 2019 *)

Formula

Equals Sum_{k >= 1} (1/2^k)*(1/5^k)*k. - Eric Desbiaux, Mar 11 2009

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

From Stefano Spezia, Jun 03 2021: (Start)

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

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