A070474 a(n) = n^3 mod 12, n^5 mod 12.

0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3, 4, 5, 0, 7, 8, 9, 4, 11, 0, 1, 8, 3

Offset: 0

Author

N. J. A. Sloane, May 12 2002

Keywords

• nonn
• easy

Comments

n^5 - n^3 == 0 (mod 12) is shown explicitly for n = 0 to 11, then the induction n -> n+12 for the 5th-order polynomial followed by binomial expansion of (n+12)^k concludes that the zero (mod 12) is periodically extended to the other integers. - R. J. Mathar, Jul 23 2009

Links

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

Crossrefs

Cf. A167176.

Programs

Magma

[Modexp(n, 3, 12 ): n in [0..100]]; // Vincenzo Librandi, Mar 27 2016

Mathematica

Table[Mod[n^3,12],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2011 *)
PowerMod[Range[0,100],3,12] (* Harvey P. Dale, Oct 29 2014 *)

PARI

a(n)=n^3%12 \\ Charles R Greathouse IV, Apr 06 2016

Sage

[power_mod(n,7,12)for n in range(0, 100)] # Zerinvary Lajos, Oct 28 2009

Formula

From R. J. Mathar, Jul 23 2009: (Start)

a(n) = a(n-12).

G.f.: -x*(1 + 8*x + 3*x^2 + 4*x^3 + 5*x^4 + 7*x^6 + 8*x^7 + 9*x^8 + 4*x^9 + 11*x^10)/ ((x-1) *(1+x+x ^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1)). (End)