A337780 a(n)/A163403(n) = the total length of the lines drawn at generation n for Conant's dissection of a square with size 1.

1, 1, 3, 5, 9, 17, 37, 73, 141, 273, 541, 1065, 2085, 4081, 8013, 15737, 30869, 60545, 118781, 233097, 457317, 897233, 1760269, 3453785, 6776181, 13294881, 26083869, 51176745, 100407301, 196998513, 386505517, 758320121, 1487807381, 4137567061

Offset

1,3

Comments

This sequence is the numerator of the fraction which gives the total length of the lines drawn at generation n for Conant's dissection of a square, assuming the square has a size of 1 unit. The corresponding denominator is given by A163403(n).

See A328078 for a description of the iterative dissection and images of the dissection after generation n.

Links

Table of n, a(n) for n=1..34.

Formula

Empirical g.f.: -(4*x^6 + 4*x^3 - 2*x^2 + 2*x + 3)/(4*x^7 - 4*x^6 + 4*x^4 - 2*x^3 + 2*x^2 + x - 1).

Example

a(1) = 1 as the first dissection line is from the bottom to the top of the square giving a total drawn length of 1/1 = 1/A163403(1) = 1.
a(2) = 1 as the second dissection line is from the left edge to halfway across the square toward the right edge, giving a total drawn length of 1/2 = 1/A163403(2) = 0.5.
a(3) = 3 as the third dissection draws two lines from the bottom edge toward the top edge, one of length 1/2 the other of length 1, giving a total drawn length of 1/2 + 1 = 3/2 = 3/A163403(3) = 1.5.
a(4) = 5 as the fourth dissection draws two lines from the left edge toward the right edge, each line being broken into two smaller lines. The total drawn length of the four smaller lines is 1/2 + 1/4 + 1/4 + 1/4 = 5/4 = 5/A163403(4) = 1.25.
a(5) = 9 as the fifth dissection draws four lines from the bottom edge toward the top edge, two lines being broken into two smaller lines. The total drawn length of the six lines is 1/4 + 1/4 + 1/2 + 1/4 + 1/4 + 3/4 = 9/4 = 9/A163403(5) = 2.25.

Crossrefs

Cf. A163403, A328078, A328080, A337642, A328081, A337675.

Sequence in context: A018095 A003217 A297011 * A178717 A006723 A217097

Adjacent sequences: A337777 A337778 A337779 * A337781 A337782 A337783

Keyword

nonn,more

Author

Scott R. Shannon, Oct 07 2020

Status

approved