A033940 a(n) = 10^n mod 7.

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

Offset: 0

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

This sequence can be employed in a test for divisibility by seven. Given the decimal expansion of some natural number, it is easily shown that the following sum has the same remainder under division by seven as the original number and that this sum is strictly smaller than the original number: Successively take the digits of the number in reverse order and multiply each of them by the respective term of the sequence A033940, then sum the products. By repeating this process, since the sums decrease in size, one ends up with seven if and only if the initial number is divisible by seven. Example: 43638 is divisible by seven since 8*1 + 3*3 + 6*2 + 3*6 + 4*4 = 63 and 3*1 + 6*3 = 21 and 1*1 + 2*3 = 7. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 16 2007

Representation of (3^n) in the circle with seven equidistant points, (10^n) mod 7=(3^n) mod 7. - Eric Desbiaux, Feb 15 2009

Representation of multiples of 3 in the circle (with seven equidistant points), see the Chryzodes links. - Eric Desbiaux, Feb 14 2009

Equivalently 3^n mod 7. - Zerinvary Lajos, Nov 24 2009

Continued fraction expansion of (269+11*sqrt(1086))/490. Decimal expansion of 1195/9009. - Klaus Brockhaus, May 24 2010

Period 6: Repeat [1, 3, 2, 6, 4, 5]. - Wesley Ivan Hurt, Jul 06 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jean-Paul Sonntag, Chryzodes "3in7"
• Jean-Paul Sonntag, Chryzodes
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Cf. A010892, A178247.

Programs

Magma

[Modexp(10, n, 7): n in [0..100]]; // Vincenzo Librandi, Feb 05 2011

Maple

A033940:=n->3^n mod 7: seq(A033940(n), n=0..100); # Wesley Ivan Hurt, Jul 05 2014

Mathematica

Table[PowerMod[10, n, 7], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Mod[3^Range[0, 100], 7] (* Wesley Ivan Hurt, Jul 06 2014 *)

PARI

a(n)=3^n%7 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[power_mod(10,n, 7)for n in range(0,106)] # Zerinvary Lajos, Nov 24 2009

Sage

[power_mod(3, n, 7)for n in range(0,106)] # Zerinvary Lajos, Nov 24 2009

Formula

a(n) = 10^n mod 7 = 3^n mod 7.

a(n) = a(n-1) - a(n-3) + a(n-4) for n>3; a(n) = a(n-6) for n>5; G.f.: (1+2*x-x^2+5*x^3)/((1-x)*(1+x)*(1-x+x^2)); a(n) = 7/2 -7*(-1)^n/6 -4*A010892(n)/3-A010892(n-1)/3. - R. J. Mathar, Feb 13 2009

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