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A001148
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Final digit of 3^n.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Let G = {1,3,7,9} ; 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. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 19 2009]
3^n mod 10 and 3^n mod 20. [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2009]
Continued fraction expansion of (243+17*sqrt(285))/4020 = 0.13183906... (see A178148) - Klaus Brockhaus, Apr 17 2011
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LINKS
| Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index to sequences with linear recurrences with constant coefficients, signature (1,-1,1).
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FORMULA
| Periodic with period 4.
a(n)= +a(n-1) -a(n-2) +a(n-3). G.f.: (1+2*x+7*x^2)/ ((1-x) * (1+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 13 2010]
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 [From Paolo P. Lava (paoloplava(AT)gmail.com), May 12 2010]
a(n) = 5-(2+I)*(-I)^n-(2-I)*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(n-2).[From Vincenzo Librandi, Feb 08 2011]
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MATHEMATICA
| Table[PowerMod[3, n, 10], {n, 0, 200}] (* From Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
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PROG
| (Sage) [power_mod(3, n, 10) for n in xrange(0, 81)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2009]
(MAGMA) [3^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011
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CROSSREFS
| Sequence in context: A096948 A179483 A016676 * A011318 A193026 A201943
Adjacent sequences: A001145 A001146 A001147 * A001149 A001150 A001151
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KEYWORD
| nonn,cofr
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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