0,2

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 stricly 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

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Feb 15 2009: Representation of (3^n) in the circle with seven equidistant points, (10^n) mod 7=(3^n) mod 7,

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

Equivalently 3^n mod 7. [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2009]

Author?, Chryzodes

Author?, Chryzodes "3in7"

Author?, Chryzodes

a(n)=a(n-1)-a(n-3)+a(n-4) = a(n-6). G.f.: (1+2x-x^2+5^x3)/((1-x)(1+x)(1-x+x^2)). a(n)=7/2 -7*(-1)^n/6 -4*A010892(n)/3-A010892(n-1)/3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009]

(Other) 1.)sage: [power_mod(10, n, 7)for n in xrange(0, 106)] # 2.)sage: [power_mod(3, n, 7)for n in xrange(0, 106)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2009]

Sequence in context: A057050 A123042 A121647 this_sequence A106409 A115510 A070264

Adjacent sequences: A033937 A033938 A033939 this_sequence A033941 A033942 A033943

nonn,new

Jeff Burch (gburch(AT)erols.com)