|
|
A033940
|
|
a(n) = 10^n mod 7.
|
|
8
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
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
Continued fraction expansion of (269+11*sqrt(1086))/490. Decimal expansion of 1195/9009. - Klaus Brockhaus, May 24 2010
|
|
LINKS
|
|
|
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
|
|
MAPLE
|
|
|
MATHEMATICA
|
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|