

A001148


Final digit of 3^n.


7



1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1
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OFFSET

0,2


COMMENTS

Let G = {1,3,7,9}, and let the binary operator o be defined as: X o Y = least significant digit of the product XY, where X,Y belong to G. Then (G,o) is an Abelian group and 3 is a generator of this group.  K.V.Iyer, Apr 19 2009
3^n mod 10 and 3^n mod 20.  Zerinvary Lajos, Nov 25 2009
Continued fraction expansion of (243+17*sqrt(285))/4020 = 0.13183906... (see A178148).  Klaus Brockhaus, Apr 17 2011


LINKS

Table of n, a(n) for n=0..80.
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1).


FORMULA

Periodic with period 4.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n1)  a(n2) + a(n3).
G.f.: (1+2*x+7*x^2)/ ((1x) * (1+x^2)). (End)
a(n) = (1/3)*(7*(n mod 4) + 4*((n+1) mod 4)  2*((n+2) mod 4) + ((n+3) mod 4)), with n>=0.  Paolo P. Lava, May 12 2010
a(n) = 5  (2+i)*(i)^n  (2i)*i^n, where i is the imaginary unit. Also a(n) = A001903(A159966(n)).  Bruno Berselli, Feb 08 2011
a(0)=1, a(1)=3, a(n) = 10  a(n2).  Vincenzo Librandi, Feb 08 2011


MATHEMATICA

Table[PowerMod[3, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)


PROG

(Sage) [power_mod(3, n, 10) for n in xrange(0, 81)] # Zerinvary Lajos, Nov 24 2009
(MAGMA) [3^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011
(PARI) a(n)=[1, 3, 9, 7][n%4+1] \\ Charles R Greathouse IV, Dec 27 2012


CROSSREFS

Sequence in context: A179483 A271879 A016676 * A262023 A275149 A011318
Adjacent sequences: A001145 A001146 A001147 * A001149 A001150 A001151


KEYWORD

nonn,cofr,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



