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A008959
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Final digit of squares: a(n) = n^2 mod 10.
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19
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0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0
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OFFSET
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0,3
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COMMENTS
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a(m*n) = a(m)*a(n) mod 10; a(5*n+k) = a(5*n-k) for k <= 5*n. - Reinhard Zumkeller, Apr 24 2009
a(n) = n^6 mod 10. - Zerinvary Lajos, Nov 06 2009
a(n) = A002015(n) mod 10 = A174452(n) mod 10. - Reinhard Zumkeller, Mar 21 2010
Decimal expansion of 166285490/1111111111. - Alexander R. Povolotsky, Mar 09 2013
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LINKS
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Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
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FORMULA
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Periodic with period 10. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = (1/5)*{(n mod 10)+2*[(n+1) mod 10]+3*[(n+2) mod 10]-[(n+3) mod 10]+[(n+5) mod 10]+2*[(n+6) mod 10]-2*[(n+7) mod 10]-[(n+8) mod 10]}. - Paolo P. Lava, Nov 24 2006
a(n) = 4.5 - (1 + 5^(1/2))*cos(Pi*n/5) + (-1 - 3/5*5^(1/2))*cos(2*Pi*n/5) + (5^(1/2) - 1)*cos(3*Pi*n/5) + (-1 + 3/5*5^(1/2))*cos(4*Pi*n/5) - 0.5*(-1)^n. - Richard Choulet, Dec 12 2008
a(n) = A010879(A000290(n)). - Reinhard Zumkeller, Jan 04 2009
G.f.: (x^9+4*x^8+9*x^7+6*x^6+5*x^5+6*x^4+9*x^3+4*x^2+x)/(-x^10+1). - Colin Barker, Aug 14 2012
a(n) = n^2 - 10*floor(n^2/10). - Wesley Ivan Hurt, Jun 12 2013
a(n) = (n - 5*A002266(n + 2))^2 + 5*(5*A002266(n + 2) mod 2). - Wesley Ivan Hurt, Jun 06 2014
a(n) = A033569(n+3) mod 10. - Wesley Ivan Hurt, Dec 06 2014
a(n) = n^k mod 10; for k > 0 where k mod 4 = 2. - Doug Bell, Jun 15 2015
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MAPLE
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A008959:=n->(n^2 mod 10); seq(A008959(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014
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MATHEMATICA
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Table[Mod[n^2, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
PowerMod[Range[0, 80], 2, 10] (* or *) LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 1, 4, 9, 6, 5, 6, 9, 4, 1}, 120] (* Harvey P. Dale, Oct 16 2012 *)
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PROG
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(Sage) [power_mod(n, 2, 10) for n in range(0, 81)] # Zerinvary Lajos, Nov 06 2009
(MAGMA) [0] cat [Intseq(n^2)[1]: n in [1..80]]; // Bruno Berselli, Feb 14 2013
(MAGMA) [n^2 - 10*Floor(n^2/10): n in [0..80]]; // Vincenzo Librandi, Jun 16 2015
(PARI) a(n)=n^2%10 \\ Charles R Greathouse IV, Sep 24 2015
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CROSSREFS
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Cf. A000290, A070431, A070435, A070438, A070442, A070452, A159852, A010879, A008960, A070514.
Sequence in context: A094090 A200632 A186723 * A316347 A169917 A059729
Adjacent sequences: A008956 A008957 A008958 * A008960 A008961 A008962
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KEYWORD
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nonn,easy,base
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AUTHOR
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N. J. A. Sloane, Mar 15 1996
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STATUS
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approved
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