A070366 a(n) = 5^n mod 9.

1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7

Offset: 0

Author

N. J. A. Sloane, May 12 2002

Keywords

• nonn
• easy

Comments

Period 6: repeat [1, 5, 7, 8, 4, 2].

Also the digital root of 5^n. - Cino Hilliard, Dec 31 2004

Digital root of the powers of any number congruent to 5 mod 9. - Alonso del Arte, Jan 26 2014

References

• Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 7, A070403; c = 8, A010689.

Cf. A000351, A007953, A010878, A010888.

Programs

Magma

[Modexp(5, n, 9): n in [0..100]]; // Wesley Ivan Hurt, Jun 28 2016

Maple

A070366:=n->power(5,n) mod 9: seq(A070366(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

Mathematica

PowerMod[5, Range[0, 120], 9] (* Harvey P. Dale, Mar 27 2011 *)
Table[Mod[5^n, 9], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)

PARI

a(n)=lift(Mod(5,9)^n); \\ Charles R Greathouse IV, Sep 24 2015

Formula

From R. J. Mathar, Apr 20 2010: (Start)

a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.

G.f.: ( 1+4*x+2*x^2+2*x^3 ) / ( (1-x)*(1+x)*(x^2-x+1) ). (End)

a(n) = 1/2^n (mod 9), n >= 0. - Wolfdieter Lang, Feb 18 2014

a(n) = A010878(A000351(n)). - Michel Marcus, Feb 20 2014

From G. C. Greubel, Mar 05 2016: (Start)

a(n) = a(n-6) for n>5.

E.g.f.: (1/2)*(9*exp(x) - exp(-x) + 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 6*exp(x/2)*cos(sqrt(3)*x/2)). (End)

a(n) = (9 - cos(n*Pi) - 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/2. - Wesley Ivan Hurt, Jun 28 2016

a(n) = 2^((-n) mod 6) mod 9. - Joe Slater, Mar 23 2017

a(n) = A007953(5*a(n-1)) = A010888(5*a(n-1)). - Stefano Spezia, Mar 20 2025