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Numbers m with A030101(m) XOR A030109(m) = m for the binary representation of m.
2

%I #28 Apr 25 2020 09:58:02

%S 1,11,91,731,5851,46811,374491,2995931,23967451,191739611,1533916891,

%T 12271335131,98170681051,785365448411,6282923587291,50263388698331,

%U 402107109586651,3216856876693211,25734855013545691,205878840108365531,1647030720866924251,13176245766935394011

%N Numbers m with A030101(m) XOR A030109(m) = m for the binary representation of m.

%C Sequence consists of all numbers with binary representation 1(011)*.

%H Colin Barker, <a href="/A245599/b245599.txt">Table of n, a(n) for n = 1..1001</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-8).

%F a(n) = 1(011)^(n-1) in binary representation.

%F a(n) = (5*8^n - 12)/28. - _Giovanni Resta_, Apr 25 2020

%F From _Colin Barker_, Apr 25 2020: (Start)

%F G.f.: x*(1 + 2*x) / ((1 - x)*(1 - 8*x)).

%F a(n) = 9*a(n-1) - 8*a(n-2) for n>2.

%F (End)

%e A030101(11) = 13, A030109(11) = 6, and 13 XOR 6 = (1101)_2 XOR (0110)_2 = (1011)_2 = 11, so 11 is in the sequence.

%t a[n_] := (5*8^n - 12)/28; Array[a, 20] (* _Giovanni Resta_, Apr 25 2020 *)

%o (PARI) Vec(x*(1 + 2*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ _Colin Barker_, Apr 25 2020

%Y Cf. A007088, A030109, A030101.

%K nonn,base,easy

%O 1,2

%A _Reinhard Muehlfeld_, Jul 27 2014

%E More terms from _Giovanni Resta_, Apr 25 2020