A343183 -id:A343183 - cp's OEIS frontend

A343182 Binary word formed from first 2^n-1 terms of paper-folding sequence A014577, reversed and complemented.

0, 100, 1100100, 110110001100100, 1101100111001000110110001100100, 110110011100100111011000110010001101100111001000110110001100100

Offset: 0

N. J. A. Sloane, May 06 2021

nonn

Take a sheet of paper, and fold the right edge up and onto the left edge. Do this n times. and unfold. Write a 0 for every valley and a 1 for every ridge, and read the sequence backwards.

a(7) is too large to include in the DATA section.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.
Sunggye Lee, Jinsoo Kim, and Won Choi, Relation between folding and un-folding paper of rectangle and (0,1)-pattern [Korean], J. Korean Soc. Math. Ed. Ser. E, 23(3) (2009), 507-522.
Rémy Sigrist and N. J. A. Sloane, Two-Dimensional Paper-Folding, Manuscript in preparation, May 2021.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]

When converted to base 10 we get A343183.

Cf. A014577. A variant of A343181.

A344022 Numbers with binary expansion (b_1, ..., b_m) such that bending a strip of paper of length k+1 with an angle of +90 degrees (resp. -90 degrees) at position X=k when b_k = 1 (resp. b_k = 0) for k = 1..m yields a configuration where all edges are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85

Offset: 1

Rémy Sigrist, May 07 2021

All positive terms belong to A166535, but the reverse is not true (for example, A166535(96) = 136 does not belong to this sequence).
This sequence is infinite as it contains A000975 and A343183.
If m belongs to the sequence, then floor(m/2) also belongs to the sequence.
For any k > 0, the sequence contains A006744(k) positive terms with k binary digits.
This sequence has connections with A258002, A255561 and A255571: these sequences encode in binary nonoverlapping or noncrossing paths in the honeycomb lattice.

			See illustration in Links section.

Rémy Sigrist, Illustration of initial terms

Cf. A000975, A006744, A166535, A258002, A255561, A255571, A343183, A344145.

is(n) = { my (b=binary(n), d=1, s=[d], z=2*d); for (k=1, #b, if (b[k], d*=I, d/=I); if (setsearch(s, z+=d), return (0), s=setunion(s, [z]); z+=d)); return (1) }

cp's OEIS Frontend

A343182 Binary word formed from first 2^n-1 terms of paper-folding sequence A014577, reversed and complemented.

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