A100601 Denominator of the best rational approximation to the decimal representation of the digital roots of m^n, m=1,2,..

9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50, 333, 1001, 100, 333, 11, 5, 9, 1001, 50

Offset

1,1

Comments

Sequence has period 9. - Nathaniel Johnston, May 05 2011

Links

Table of n, a(n) for n=1..57.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Formula

The digital root of a number m > 0, is d = m mod 9 if d > 0 else d = 9.

G.f.: -x*(5*x^8+11*x^7+333*x^6+100*x^5+1001*x^4+333*x^3+50*x^2+1001*x+9) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Jun 23 2014

a(n) = a(n-9). - Wesley Ivan Hurt, Oct 18 2021

Example

Digital roots of 4^n = 1,4,7,1,4,7,1,4,7,1,4,7,.. 49/333 = 0.147147147147147147147147147147147,.. 333 is the 4th term in the sequence.

Prog

(PARI) f(n, m) = for(x=0, n, print1(droot(m^x)", ")) droot(n) = \ the digital root of a number. { local(x); x= n%9; if(x>0, return(x), return(9)) }
(PARI) Vec(-x*(5*x^8+11*x^7+333*x^6+100*x^5+1001*x^4+333*x^3+50*x^2+1001*x+9) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

Crossrefs

Cf. A100406, A100579.

Sequence in context: A307324 A327435 A350967 * A260029 A266321 A228293

Adjacent sequences: A100598 A100599 A100600 * A100602 A100603 A100604

Keyword

base,frac,easy,nonn

Author

Cino Hilliard, Jan 02 2005

Extensions

Offset corrected by Nathaniel Johnston, May 05 2011

Status

approved