A155754 A variation on 10^n mod 19

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

Offset

0,2

Comments

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

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.

The test is successful if the final sum is 0.

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

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

Links

Table of n, a(n) for n=0..71.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,-1).

Formula

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]

Crossrefs

Cf. A033940, A119910, A117378.

Sequence in context: A011259 A376330 A377616 * A273840 A117019 A155692

Adjacent sequences: A155751 A155752 A155753 * A155755 A155756 A155757

Keyword

easy,sign

Author

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

Status

approved