The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A327232 Smallest integer k such that the set of all n consecutive digits of k equals the set of 0 to 2^n-1 written as n-digit binary numbers. 2
 10, 10011, 1000101110, 1000010011010111100, 100000100011001010011101011011111000, 100000010000110001010001110010010110011010011110101011101101111110000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS floor(a(n)/10^(n-1)) is the juxtaposition of a de Bruijn sequence. [This is because the first and last n-1 digits of a(n) are always identical - see my link for a general proof. - Jianing Song, Oct 29 2019] LINKS Eric Weisstein's World of Mathematics, de Bruijn Sequence Wikipedia, de Bruijn Sequence FORMULA a(n) = A007088(A166315(n))*10^(n-2) + 10^(2^n+n-2) for n > 1. Proof: by the property mentioned in the comment section, write a(n) = (d_1)*10^(2^n+n-2) + (d_2)*10^(2^n+n-3) + ... + (d_2^n)*10^(n-1) + (d_1)*10^(n-2) + (d_2)*10^(n-3) + ... + (d_(n-1))*10^0, d_i = 0 or 1, then d_1 = 1, (d_2)*2^(2^n-1) + (d_3)*2^(2^n-2) + ... + (d_2^n)*2^1 + (d_1)*2^0 >= A166315(n), and d_2, d_3, ..., d_(n-1) >= 0. The equalities can hold simultaneously (when written as a 2^n-digit binary number, A166315(n) begins with n 0's and ends with a 1), which gives the formula. - Jianing Song, Oct 28 2019 a(n) = A004086(floor(A327233(n)/10^(n-2)))*10^(n-2) for n > 1. - Jinyuan Wang, Nov 02 2019 EXAMPLE For n = 2, the set of all n consecutive digits of 10011 is {10, 00, 01, 11} and the set of 0 to 2^n-1 in binary is {00, 01, 10, 11}. For n = 3, the set of all n consecutive digits of 1000101110 is {100, 000, 001, 010, 101, 011, 111, 110} and the set of 0 to 2^n-1 in binary is {000, 001, 010, 011, 100, 101, 110, 111}. PROG (PARI) a(n) = {my(v=vector(2^n), w=vector(2^n)); k=2^(2^n+n-2)-1; for(i=1, 2^n, v[i]=fromdigits(binary(i-1))); while(Set(w)!=v, u=binary(k++); w[1]=fromdigits(u)\10^(2^n-1); for(i=2, 2^n, w[i]=u[i+n-1]+10*(w[i-1]%10^(n-1)))); fromdigits(u); } CROSSREFS Cf. A007088, A166315 (earliest binary de Bruijn sequences), A166316 (largest binary de Bruijn sequences), A327233 (largest k). Sequence in context: A139109 A317959 A119037 * A266841 A267677 A101305 Adjacent sequences: A327229 A327230 A327231 * A327233 A327234 A327235 KEYWORD nonn AUTHOR Jinyuan Wang, Oct 26 2019 EXTENSIONS a(5)-a(6) from Jinyuan Wang, Nov 02 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)