A113217 Parity of decimal digital root of n.

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1

Offset: 0

Author

Reinhard Zumkeller, Oct 18 2005

Keywords

• nonn
• base
• easy

Comments

Except for the first element, the sequence is periodic (with a period of length 9). The sequence corresponds to that produced by a prescribed set of bitwise operations. The (sub)sequence is produced starting from input pairs (0,1),(1,1),(1,0). For example, (0,1) acted on (in succession) by [and,xor,or,xor,or,and,or,and,xor], with the same operation set then repeated. For clarity, the example is AND(0,1) is 0. XOR(1,0) is 1. OR(0,1) is 1. XOR(1,1) is 0. OR(1,0) is 1. AND(0,1) is 0. OR(1,0) is 1. AND(0,1) is 0. XOR(1,0) is 1. Repeat. The analysis was done using Gnumeric's built-in functions. In this example, the inputs align to n=2,3, and the operation results to the next 7 elements. The (3) starting input pairs mentioned begin at bitwise operator positions 1,2 and 5. - Bill McEachen, May 24 2014

Links

• Eric Weisstein's World of Mathematics, Digital Root
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A000035, A010888, A113218.

Programs

Mathematica

Table[Mod[ResourceFunction["AdditiveDigitalRoot"][n],2],{n,0,104}] (* James C. McMahon, Jun 19 2024 *)

Formula

a(n) = A010888(n) mod 2.

a(n) = if n mod 9 = 1 then 1 else 1 - a(n-1), a(0)=0.

a(n) = A000035(A010888(n)). - Omar E. Pol, Oct 28 2013

a(n) = (1+(-1)^floor(8*n/9))/2 for n>0. - Wesley Ivan Hurt, Apr 27 2020