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A113217 Parity of decimal digital root of n.

%I A113217 #36 Feb 16 2025 08:32:59
%S A113217 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,
%T A113217 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,
%U A113217 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
%N A113217 Parity of decimal digital root of n.
%C A113217 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
%H A113217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>
%H A113217 <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 A113217 a(n) = A010888(n) mod 2.
%F A113217 a(n) = if n mod 9 = 1 then 1 else 1 - a(n-1), a(0)=0.
%F A113217 a(n) = A000035(A010888(n)). - _Omar E. Pol_, Oct 28 2013
%F A113217 a(n) = (1+(-1)^floor(8*n/9))/2 for n>0. - _Wesley Ivan Hurt_, Apr 27 2020
%t A113217 Table[Mod[ResourceFunction["AdditiveDigitalRoot"][n],2],{n,0,104}] (* _James C. McMahon_, Jun 19 2024 *)
%Y A113217 Cf. A000035, A010888, A113218.
%K A113217 nonn,base,easy
%O A113217 0,1
%A A113217 _Reinhard Zumkeller_, Oct 18 2005