cp's OEIS Frontend

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.

%I A337780 #37 May 23 2021 00:20:55
%S A337780 1,1,3,5,9,17,37,73,141,273,541,1065,2085,4081,8013,15737,30869,60545,
%T A337780 118781,233097,457317,897233,1760269,3453785,6776181,13294881,
%U A337780 26083869,51176745,100407301,196998513,386505517,758320121,1487807381,4137567061
%N 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.
%C A337780 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).
%C A337780 See A328078 for a description of the iterative dissection and images of the dissection after generation n.
%F A337780 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).
%e A337780 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.
%e A337780 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.
%e A337780 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.
%e A337780 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.
%e A337780 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.
%Y A337780 Cf. A163403, A328078, A328080, A337642, A328081, A337675.
%K A337780 nonn,more
%O A337780 1,3
%A A337780 _Scott R. Shannon_, Oct 07 2020