A204688 a(n) = n^n (mod 3).

1, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1

Offset: 0

Author

José María Grau Ribas, Jan 18 2012

Keywords

• nonn
• easy

Comments

For n>0, a(n) is periodic with period 6: repeat [1, 1, 0, 1, 2, 0].

Decimal expansion of 1110119/9999990. - David A. Corneth, Jun 28 2016

Links

• José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv preprint arXiv:1203.4066 [math.NT], 2012.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Cf. A000312, A010872, A174824.

Programs

Magma

[1] cat [Modexp(n, n, 3): n in [1..100]]; // Wesley Ivan Hurt, Jun 28 2016

Maple

A204688:=n->power(n,n) mod 3: 1, seq(A204688(n), n=1..100); # Wesley Ivan Hurt, Jun 28 2016

Mathematica

Table[PowerMod[n,n,3], {n,0,140}]

Formula

G.f.: (1+x+x^2+x^4+2*x^5-x^6) / (1-x^6). - Bruno Berselli, Jan 18 2012

From Wesley Ivan Hurt, Jun 28 2016: (Start)

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

a(n) = sin(n*Pi/3) * (10*sin(n*Pi/3) + 2*sin(2*n*Pi/3) - sqrt(3) - 2*sqrt(3)*cos(n*Pi/3))/6 for n>0. (End)

a(n) = A010872(A000312(n)). - Michel Marcus, Jun 28 2016