A008960 Final digit of cubes: n^3 mod 10.

0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 0, 1, 8, 7, 4, 5

Offset

0,3

Comments

Decimal expansion of 208284810/1111111111. - Alexander R. Povolotsky, Mar 08 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

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) = 4.5 -cos(Pi*n/5) +(1/2*(-(5-5^(1/2))^(1/2) +(5+5^(1/2))^(1/2))*2^(1/2))*sin(Pi*n/5) -cos(2*Pi*n/5) +(-1/10*(-(5-5^(1/2))^(1/2)+3*(5+5^(1/2))^(1/2))*2^(1/2))*sin(2*Pi*n/5) -cos(3*Pi*n/5) +(-1/2*((5-5^(1/2))^(1/2) +(5+5^(1/2))^(1/2))*2^(1/2))*sin(3*Pi*n/5) -cos(4*Pi*n/5) +( -1/10*(3*(5-5^(1/2))^(1/2) +(5 +5^(1/2))^(1/2))*2^(1/2))*sin(4*Pi*n/5) -0.5*(-1)^n. - Richard Choulet, Dec 12 2008

a(n) = n^k mod 10; for k > 0 where k mod 4 = 3. - Doug Bell, Jun 15 2015

G.f.: x*(1+8*x+7*x^2+4*x^3+5*x^4+6*x^5+3*x^6+2*x^7+9*x^8) / ((1-x)*(1+x)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)). - Colin Barker, Nov 30 2015

Mathematica

Table[Mod[n^3, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2011 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 1, 8, 7, 4, 5, 6, 3, 2, 9}, 81] (* Ray Chandler, Aug 26 2015 *)

Prog

(Sage) [power_mod(n, 3, 10 ) for n in range(0, 81)] # Zerinvary Lajos, Oct 29 2009
(PARI) a(n)=n^3%10 \\ Charles R Greathouse IV, Mar 08 2013
(Magma) [n^3 mod 10: n in [0..80]]; // Vincenzo Librandi, Mar 26 2013
(PARI) concat(0, Vec(x*(1+8*x+7*x^2+4*x^3+5*x^4+6*x^5+3*x^6+2*x^7+9*x^8) / ((1-x)*(1+x)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)) + O(x^100))) \\ Colin Barker, Nov 30 2015

Crossrefs

Cf. A167176.

Cf. A010879, A008959, A070514. - Doug Bell, Jun 15 2015

Sequence in context: A249136 A154815 A085848 * A077744 A111448 A169885

Adjacent sequences: A008957 A008958 A008959 * A008961 A008962 A008963

Keyword

nonn,easy,base

Author

N. J. A. Sloane

Status

approved