login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A100406 a(n) = repeating period of the digital roots of the sequence {m^n, m=1,2,3...}. 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 (list; graph; refs; listen; history; text; internal format)
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

a(n) = (1/81)*(70843*(n mod 9) + 70852*((n+1) mod 9) + 72175*((n+2) mod 9) + 69286*((n+3) mod 9) + 1491268*((n+4) mod 9) - 1348484*((n+5) mod 9) + 69529*((n+6) mod 9) + 1194565*((n+7) mod 9) - 1053095*((n+8) mod 9)), with n>=0. - Paolo P. Lava, Oct 21 2008

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 *)

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: A030639 A261439 A182658 * A183797 A234783 A206134

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 16:27 EST 2016. Contains 278946 sequences.