

A327233


Largest integer k < 10^(2^n+n) such that the set of all n consecutive digits of k equals the set of 0 to 2^n1 written as ndigit binary numbers.


1




OFFSET

1,1


COMMENTS

floor(a(n)/10^(n1)) is the juxtaposition of a de Bruijn sequence. [This is because the first and last n1 digits of a(n) are always identical  see my link for a general proof.  Jianing Song, Oct 29 2019]


LINKS

Table of n, a(n) for n=1..6.
Jianing Song, A general proof that the first and last n1 digits of a(n) are identical
Eric Weisstein's World of Mathematics, de Bruijn Sequence
Wikipedia, de Bruijn Sequence


FORMULA

a(n) = A004086(A327232(n))*10^(n2) + A002275(n2) for n > 1.
a(n) = A007088(A166316(n))*10^(n1) + A002275(n1).
Proof: by the property mentioned in the comment section, write a(n) = (d_1)*10^(2^n+n2) + (d_2)*10^(2^n+n3) + ... + (d_2^n)*10^(n1) + (d_1)*10^(n2) + (d_2)*10^(n3) + ... + (d_(n1))*10^0, d_i = 0 or 1, then (d_1)*2^(2^n1) + (d_2)*2^(2^n2) + ... + (d_2^n)*2^0 <= A166316(n), and d_1, d_2, ..., d_(n1) <= 1. The equalities can hold simultaneously (when written as a 2^ndigit binary number, A166316(n) begins with n 1's), which gives the formula.  Jianing Song, Oct 28 2019


CROSSREFS

Cf. A007088, A166315 (earliest binary de Bruijn sequences), A166316 (largest binary de Bruijn sequences), A327232 (smallest k).
Sequence in context: A267705 A086164 A138119 * A138118 A267130 A267850
Adjacent sequences: A327230 A327231 A327232 * A327234 A327235 A327236


KEYWORD

nonn


AUTHOR

Jinyuan Wang, Oct 26 2019


STATUS

approved



