A070365 a(n) = 5^n mod 7.

1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2

Offset: 0

Author

N. J. A. Sloane, May 12 2002

Keywords

• nonn
• easy

Comments

From Klaus Brockhaus, May 23 2010: (Start)

Period 6: repeat [1, 5, 4, 6, 2, 3].

Continued fraction expansion of (221+11*sqrt(1086))/490.

Decimal expansion of 199/1287.

First bisection is A153727. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Cf. A178229 (decimal expansion of (221+11*sqrt(1086))/490), A178141 (repeat 4, -1, 2, -4, 1, -2), A117373 (repeat 1, -2, -3, -1, 2, 3), A153727 (trajectory of 3x+1 sequence starting at 1).

Cf. A000351, A010876.

Programs

Magma

[Modexp(5, n, 7): n in [0..100]]; // Vincenzo Librandi, Mar 24 2016 - after Bruno Berselli

Maple

A070365:=n->[1, 5, 4, 6, 2, 3][(n mod 6)+1]: seq(A070365(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

Mathematica

PowerMod[5, Range[0, 110], 7] (* or *) LinearRecurrence[{1, 0, -1, 1}, {1, 5, 4, 6}, 110] (* Harvey P. Dale, Apr 26 2011 *)
Table[Mod[5^n, 7], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)
PadRight[{}, 100, {1, 5, 4, 6, 2, 3}] (* or *) CoefficientList[Series[(1 + 5 x + 4 x^2 + 6 x^3 + 2 x^4 + 3 x^5) / (1 - x^6), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 24 2016 *)

PARI

a(n)=lift(Mod(5,7)^n) \\ Charles R Greathouse IV, Mar 22 2016

Formula

From R. J. Mathar, Apr 13 2010: (Start)

a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.

G.f.: (1+4*x-x^2+3*x^3)/ ((1-x)*(1+x)*(x^2-x+1)). (End)

From Klaus Brockhaus, May 23 2010: (Start)

a(n+1)-a(n) = A178141(n).

a(n+2)-a(n) = A117373(n+5). (End)

From G. C. Greubel, Mar 05 2016: (Start)

a(n) = a(n-6) for n>5.

E.g.f.: (1/3)*(7*cosh(x) + 14*sinh(x) + 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 4*exp(x/2)*cos(sqrt(3)*x/2)). (End)

a(n) = (21 - 7*cos(n*Pi) - 8*cos(n*Pi/3) + 4*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 23 2016

a(n) = A010876(A000351(n)). - Michel Marcus, Jun 27 2016