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 A000689 Final decimal digit of 2^n. 10
 1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS These are the analogs of the powers of 2 in carryless arithmetic mod 10. Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (Also see the first comment under A001148). - K.V.Iyer, Mar 12 2010 a(n) is also 2^n mod 10. For n > 0: a(n) = A002081(n) - A002081(n-1). LINKS David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version Index entries for linear recurrences with constant coefficients, signature (1, -1, 1). FORMULA Periodic with period 4. a(n) = (1/6)*{8*(n mod 4)-[(n+1) mod 4]+2*[(n+2) mod 4]+11*[(n+3) mod 4]}-5*{1-[((n+1)!+1) mod (n+1)]}, with n>=0. - Paolo P. Lava, Jun 25 2007; corrected by Paolo P. Lava, Mar 23 2010 a(n) = +a(n-1) -a(n-2) +a(n-3), n>3. G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). - R. J. Mathar, Apr 13 2010 a(n) = 5+(1/2)*[(1+3*I)*I^n+(1-3*I)*(-I)^n]-5*[C(2*n,n) mod 2], with n>=0. - Paolo P. Lava, May 10 2010 For n>=1, a(n) = 10 - (4x^3 +47x -27x^2)/3, where x = (n+3) mod 4 + 1. For n>=1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ). G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012 EXAMPLE G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ... MAPLE P:=proc(n) local a, i; for i from 0 by 1 to n do a:=1/6*(8*(i mod 4)-((i+1) mod 4)+2*((i+2) mod 4)+11*((i+3) mod 4))-5*(1-(((i+1)!+1) mod (i+1))); print(a); od; end: P(100); # Paolo P. Lava, Jun 25 2007 MATHEMATICA Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *) PROG (Sage) [power_mod(2, n, 10)for n in range(0, 81)] #  Zerinvary Lajos, Nov 03 2009 (PARI) for(n=0, 80, if(n, {x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3, ", ")}, {print1("1, ")})) (MAGMA) [2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011 (Haskell) a000689 n = a000689_list !! n a000689_list = 1 : cycle [2, 4, 8, 6]  -- Reinhard Zumkeller, Sep 15 2011 CROSSREFS Cf. A173635. Sequence in context: A125733 A333555 A280426 * A132137 A011180 A103546 Adjacent sequences:  A000686 A000687 A000688 * A000690 A000691 A000692 KEYWORD nonn,base,easy AUTHOR STATUS approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)