A076309 a(n) = floor(n/10) - 2*(n mod 10).

0, -2, -4, -6, -8, -10, -12, -14, -16, -18, 1, -1, -3, -5, -7, -9, -11, -13, -15, -17, 2, 0, -2, -4, -6, -8, -10, -12, -14, -16, 3, 1, -1, -3, -5, -7, -9, -11, -13, -15, 4, 2, 0, -2, -4, -6, -8, -10, -12, -14, 5, 3, 1, -1, -3, -5, -7, -9, -11, -13, 6, 4, 2, 0, -2, -4, -6, -8, -10

Offset: 0

Author

Reinhard Zumkeller, Oct 06 2002

Keywords

• sign
• base
• easy

Comments

Delete the last digit from n and subtract twice this digit from the shortened number. - N. J. A. Sloane, May 25 2019

(n==0 modulo 7) iff (a(n)==0 modulo 7); applied recursively, this property provides a useful test for divisibility by 7.

Examples

			695591 is not a multiple of 7, as 695591 -> 69559-2*1=69557 -> 6955-2*7=6941 -> 694-2*1=692 -> 69-2*2=65=7*9+2, therefore the answer is NO.
Is 3206 divisible by 7? 3206 -> 320-2*6=308 -> 30-2*8=14=7*2, therefore the answer is YES, indeed 3206=2*7*229.
		

References

• Paul Erdős and János Surányi. Topics in the Theory of Numbers. New York: Springer, 2003. Problem 6, page 3.
• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.2 Fundamental Operations, p. 121.
• Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Divisibility Tests.
• Wikipedia, Divisibility rule.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A008589, A076310, A076311, A076312.

Programs

Haskell

a076309 n =  n' - 2 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

Mathematica

Table[Floor[n/10] - 2*Mod[n, 10], {n, 0, 100}] (* G. C. Greubel, Apr 07 2016 *)

PARI

a(n) = n\10 - 2*(n % 10); \\ Michel Marcus, Apr 07 2016

Formula

From R. J. Mathar, Nov 23 2010: (Start)

a(n) = a(n-1) + a(n-10) - a(n-11).

G.f.: x*(-2 -2*x -2*x^2 -2*x^3 -2*x^4 -2*x^5 -2*x^6 -2*x^7 -2*x^8 +19*x^9)/((1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^2). (End)