A100406 a(n) = repeating period of the digital roots of the sequence {m^n, m=1,2,3...}.

1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1, 124875, 9, 147, 157842, 9, 174, 18, 9, 1

Offset

1,2

Comments

Sequence has period 9.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Formula

G.f.: -x*(9*x^8+18*x^7+174*x^6+9*x^5+157842*x^4+147*x^3+9*x^2+124875*x+1) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Jun 25 2014

Example

The digital roots of 1^n are 1,1,1,1,1,1,.. so 1 is the repeating decimal period for 1^n.
The digital roots of 2^n are 1,2,4,8,7,5.. so 124875 is the repeating decimal period for 2^n.
The digital roots of 3^n are 1,3,9,9,9,9,.. so 9 is the repeating decimal period for 3^n.

Mathematica

CoefficientList[Series[-(9 x^8 + 18 x^7 + 174 x^6 + 9 x^5 + 157842 x^4 + 147 x^3 + 9 x^2 + 124875 x + 1)/((x - 1) (x^2 + x + 1) (x^6 + x^3 + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 26 2014 *)
PadRight[{}, 100, {1, 124875, 9, 147, 157842, 9, 174, 18, 9}] (* Harvey P. Dale, Mar 29 2019 *)

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*(9*x^8+18*x^7+174*x^6+9*x^5+157842*x^4+147*x^3+9*x^2+124875*x+1) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Jun 25 2014

Crossrefs

Cf. A100579, A100601.

Sequence in context: A261439 A182658 A282919 * A351948 A341115 A183797

Adjacent sequences: A100403 A100404 A100405 * A100407 A100408 A100409

Keyword

easy,nonn,base

Author

Cino Hilliard, Dec 31 2004

Extensions

Offset corrected by Nathaniel Johnston, May 05 2011

Status

approved