A230598 Lexicographically earliest sequence of distinct positive integers such that all black pixels in the binary plot of the sequence are connected (see Comments for details).

1, 3, 2, 6, 4, 5, 7, 9, 11, 10, 14, 8, 12, 13, 15, 17, 19, 18, 22, 20, 21, 23, 25, 27, 26, 30, 16, 24, 28, 29, 31, 33, 35, 34, 38, 36, 37, 39, 41, 43, 42, 46, 40, 44, 45, 47, 49, 51, 50, 54, 52, 53, 55, 57, 59, 58, 62, 32, 48, 56, 60, 61, 63, 65, 67, 66, 70

Offset

1,2

Comments

For any n, m, i, j such that a(n) AND (2^i) <> 0, and a(m) AND (2^j) <>0 (where AND stands for the bitwise AND operator), there exist two sequences of finite length L, say p and b, such that:

(1) p(1)=n, b(1)=i,

(2) p(L)=m, b(L)=j,

(3) a(p(k)) AND (2^b(k)) <> 0 for any k between 1 and L,

(4) |p(k+1)-p(k)| + |b(k+1)-b(k)| = 1 for any k between 1 and L-1.

These two finite sequences define a path of black pixels connecting the black pixels at positions (n,i) and (m,j).

Links

Paul Tek, Table of n, a(n) for n = 1..10000

Paul Tek, Binary plot of the first 127 terms

Paul Tek, PERL program for this sequence

Eric Weisstein's World of Mathematics, Binary Plot

Index entries for sequences that are permutations of the natural numbers

Formula

Empirically, for any k>2 :

(1) a(2^k-1) = 2^k-1,

(2) a(2^k) = 2^k+1,

(3) a(n) = a(n-2^k+1) + 2^k, for any n such that 2^k<=n<2^(k+1)-(k+1),

(4) a(n) = 2^k, for n=2^(k+1)-(k+1),

(5) a(n) = a(n-2^k) + 2^k, for any n such that 2^(k+1)-(k+1)<n<2^(k+1).

Prog

(Perl) See Link section.

Crossrefs

Cf. A226077, A230599.

Sequence in context: A196047 A106409 A115510 * A245446 A070264 A365233

Adjacent sequences: A230595 A230596 A230597 * A230599 A230600 A230601

Keyword

nonn,base,nice

Author

Paul Tek, Oct 24 2013

Status

approved