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A008959 Final digit of squares: n^2 mod 10. 18
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

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).

FORMULA

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

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 xrange(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

Cf. A000290, A070431, A070435, A070438, A070442, A070452, A159852, A010879, A008960, A070514.

Sequence in context: A094090 A200632 A186723 * A169917 A059729 A184988

Adjacent sequences:  A008956 A008957 A008958 * A008960 A008961 A008962

KEYWORD

nonn,easy,base

AUTHOR

N. J. A. Sloane, Mar 15 1996

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.