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A083655 Numbers which do not appear prematurely in the binary Champernowne word (A030190).

%I A083655 #55 Dec 17 2022 08:26:26
%S A083655 0,1,2,4,8,10,16,32,36,64,128,136,256,512,528,1024,2048,2080,4096,
%T A083655 8192,8256,16384,32768,32896,65536,131072,131328,262144,524288,524800,
%U A083655 1048576,2097152,2098176,4194304,8388608,8390656,16777216,33554432
%N A083655 Numbers which do not appear prematurely in the binary Champernowne word (A030190).
%C A083655 In other words, numbers k whose binary expansion first appears in A030190 at its expected place, i.e., n appears first starting at position A296349(n). - _N. J. A. Sloane_, Dec 17 2017
%C A083655 a(n) are the Base 2 "Punctual Bird" numbers: write the nonnegative integers, base 2, in a string 011011100101110111.... Sequence gives numbers which do not occur in the string ahead of their natural place. - _Graeme McRae_, Aug 11 2007
%H A083655 Graeme McRae, Aug 11 2007, <a href="/A083655/b083655.txt">Table of n, a(n) for n = 0..113</a> - Corrected by _Rémy Sigrist_, Jun 14 2020
%H A083655 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-8).
%F A083655 A083653(a(n))=a(n), A083654(a(n))=1.
%F A083655 a(0)=0, a(1)=1, a(2)=2; then for n>=1, a(3n)=2^(2n), a(3n+1)=2^(2n+1), a(3n+2)=2^(2n+1)+2^n. - _Graeme McRae_, Aug 11 2007
%F A083655 From _Colin Barker_, Jun 14 2020: (Start)
%F A083655 G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 - 2*x^4 - 8*x^5 - 8*x^6 - 8*x^7) / ((1 - 2*x^3)*(1 - 4*x^3)).
%F A083655 a(n) = 6*a(n-3) - 8*a(n-6) for n>8. (End)
%F A083655 a(n) = 2^floor(2*(n+2)/3-1) + (floor((n+1)/3)-floor(n/3))*2^(floor(n/3)) - floor(5/(n+3)). - _Alan Michael Gómez Calderón_, Dec 15 2022
%t A083655 LinearRecurrence[{0,0,6,0,0,-8},{0,1,2,4,8,10,16,32,36},50] (* _Harvey P. Dale_, Aug 19 2020 *)
%o A083655 (PARI) a(n)= if (n<=2, n, my (m=n\3); if (n%3==0, 2^(2*m), n%3==1, 2^(2*m+1), 2^m + 2^(2*m+1)))  \\ _Rémy Sigrist_, Jun 14 2020
%o A083655 (PARI) concat(0, Vec(x*(1 + 2*x + 4*x^2 + 2*x^3 - 2*x^4 - 8*x^5 - 8*x^6 - 8*x^7) / ((1 - 2*x^3)*(1 - 4*x^3)) + O(x^40))) \\ _Colin Barker_, Jun 14 2020
%o A083655 (PARI) a(n) = 2^((2*n+1)\3) + (n%3==2)<<(n\3) - (n<3) \\ _Charles R Greathouse IV_, Dec 16 2022
%Y A083655 A000079 is a subsequence.
%Y A083655 Cf. A030190, A030304, A007088, A131881, A132131.
%Y A083655 For the complement, the "Early Bird" numbers, see A296365.
%K A083655 nonn,base,easy
%O A083655 0,3
%A A083655 _Reinhard Zumkeller_, May 01 2003
%E A083655 More terms from _Graeme McRae_, Aug 11 2007