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A038713 a(n) = n XOR (n-1), i.e., nim-sum of sequential pairs, written in binary.

%I A038713 #30 Sep 21 2023 01:45:14
%S A038713 1,11,1,111,1,11,1,1111,1,11,1,111,1,11,1,11111,1,11,1,111,1,11,1,
%T A038713 1111,1,11,1,111,1,11,1,111111,1,11,1,111,1,11,1,1111,1,11,1,111,1,11,
%U A038713 1,11111,1,11,1,111,1,11,1,1111,1,11,1,111,1,11,1,1111111,1,11,1,111,1
%N A038713 a(n) = n XOR (n-1), i.e., nim-sum of sequential pairs, written in binary.
%H A038713 Vincenzo Librandi, <a href="/A038713/b038713.txt">Table of n, a(n) for n = 1..1000</a>
%H A038713 <a href="/index/Ni#Nimsums">Index entries for sequences related to Nim-sums</a>
%F A038713 a(n) = (10^A001511(n) - 1)/(10 - 1).
%F A038713 Multiplicative with a(2^e) = (10^(e+1) - 1)/9, a(p^e) = 1 if p odd.
%F A038713 G.f.: Sum_{k>=0} 10^k * x^(2^k) / (1 - x^(2^k)). - _Ilya Gutkovskiy_, Dec 15 2020
%F A038713 Dirichlet g.f.: zeta(s) * 2^s/(2^s - 10). - _Amiram Eldar_, Sep 21 2023
%e A038713 a(6) = 11 because 110 XOR 101 = 11 base 2.
%t A038713 Table[(10^IntegerExponent[2*n, 2] - 1)/9, {n, 100}] (* _Vincenzo Librandi_, Mar 11 2013 *)
%o A038713 (PARI) a(n)=if(n<1,0, (10^(valuation(n,2)+1)-1)/9) /* _Michael Somos_, Apr 28 2005 */
%o A038713 (Magma) [(10^(Valuation(n, 2)+1)-1)/9: n in [1..70]]; // _Vincenzo Librandi_, Mar 11 2013
%Y A038713 Cf. A038712 translated to binary, A001511 translated to unary (repeated 1's).
%K A038713 mult,easy,nonn
%O A038713 1,2
%A A038713 _Henry Bottomley_, May 02 2000