A155754 - OEIS

%I #13 May 06 2024 06:50:09

%S 1,-9,5,-7,6,3,-8,-4,-2,-1,9,-5,7,-6,-3,8,4,2,1,-9,5,-7,6,3,-8,-4,-2,

%T -1,9,-5,7,-6,-3,8,4,2,1,-9,5,-7,6,3,-8,-4,-2,-1,9,-5,7,-6,-3,8,4,2,1,

%U -9,5,-7,6,3,-8,-4,-2,-1,9,-5,7,-6,-3,8,4,2

%N A variation on 10^n mod 19

%C This sequence can be employed in a test for divisibility by 19 and works like A033940 works for 7.

%C The use of negative coefficients ensures the termination of the test because the modulus of the intermediate sum at each step of the test decreases strictly.

%C The test is successful if the final sum is 0.

%C The negative coefficients have the form (10^n mod 19) - 19 when 10^n mod 19 > 9.

%C Example: 8284 is divisible by 19 since |4*1 + 8*(-9) + 2*5 + 8*(-7)| = 114 and 4*1 + 1*(-9) + 1*5 = 0.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,-1).

%F a(n) = -a(n-9). G.f.: (-2*x^8-4*x^7-8*x^6+3*x^5+6*x^4-7*x^3+5*x^2-9*x+1) / (x^9+1). [_Colin Barker_, Feb 14 2013]

%Y Cf. A033940, A119910, A117378.

%K easy,sign

%O 0,2

%A Ferruccio Guidi (fguidi(AT)cs.unibo.it), Jan 26 2009