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

0, 1, 2, 4, 8, 10, 16, 32, 36, 64, 128, 136, 256, 512, 528, 1024, 2048, 2080, 4096, 8192, 8256, 16384, 32768, 32896, 65536, 131072, 131328, 262144, 524288, 524800, 1048576, 2097152, 2098176, 4194304, 8388608, 8390656, 16777216, 33554432

Offset: 0

Author

Reinhard Zumkeller, May 01 2003

Keywords

• nonn
• base
• easy

Comments

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

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

Links

• Graeme McRae, Aug 11 2007, Table of n, a(n) for n = 0..113 - Corrected by _Rémy Sigrist_, Jun 14 2020
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-8).

Crossrefs

A000079 is a subsequence.

Cf. A030190, A030304, A007088, A131881, A132131.

For the complement, the "Early Bird" numbers, see A296365.

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-8},{0,1,2,4,8,10,16,32,36},50] (* Harvey P. Dale, Aug 19 2020 *)

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

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

PARI

a(n) = 2^((2*n+1)\3) + (n%3==2)<<(n\3) - (n<3) \\ Charles R Greathouse IV, Dec 16 2022

Formula

A083653(a(n))=a(n), A083654(a(n))=1.

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

From Colin Barker, Jun 14 2020: (Start)

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)).

a(n) = 6*a(n-3) - 8*a(n-6) for n>8. (End)

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

Extensions

More terms from Graeme McRae, Aug 11 2007