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A008959 Final digit of squares: a(n) = n^2 mod 10. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
FORMULA
Periodic with period 10. - Franklin T. Adams-Watters, Mar 13 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
MAPLE
A008959:=n->(n^2 mod 10); seq(A008959(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A094090 A200632 A186723 * A316347 A169917 A059729
KEYWORD
nonn,easy,base
AUTHOR
N. J. A. Sloane, Mar 15 1996
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)