OEIS/Attaching segments

Column 5 (contracting)

attach to: 3,7,11,15,19 ... => 3 mod 4
left side: 1,2,3,4,5,6 ...  => 1 mod 1
remaining:                 0,1,2 mod 4

Column 6 (contracting)

attach to: 1,5,9,13,17 ...  => 1 mod 4
left side: 1,4,7,10,13         1 mod 3
remaining:                   0,2 mod 4
double steps: 
  if (5) 3 mod 4 = LS(6) 1 mod 3: 7,19,31 => 7 mod 12
  if (6) 1 mod 4 = LS(6) 1 mod 3: 1,13,25 => 1 mod 12

Column 9 (contracting)

attach to: 2,10,18,26,34    => 2 mod 8
left side: 1,4,7,10,13         1 mod 3 
remaining:                 0,4,6 mod 8
double steps:
  if (9) 2 mod 8 = LS(6) 1 mod 3: 10,34,58 => 10 mod 24

Column 10 (expanding)

attach to: 6,14,22          => 6 mod 8
left side: 7,16,25             7 mod 9 
remaining:                   0,4 mod 8
double steps:
  if (10) 6 mod 8 = LS(6)  1 mod 3: 22,46,70 => 22 mod 24
  if (6)  1 mod 4 = LS(10) 7 mod 9: 25,61,97 => 25 mod 36
  if (5)  3 mod 4 = LS(10) 7 mod 9: 7,43,79  =>  7 mod 36
triple steps:
  if 1 mod 12 = 25 mod 36        => 25 mod 36
  if 7 mod 12 =  7 mod 36        =>  7 mod 36

Column 13 (contracting)

attach to: 12,28,44              => 12 mod 16
left side: 7,16,25                   7 mod 9 
remaining:                       0,4,8 mod 16
double steps:
  if 12 mod 16 = 1 mod 3: 28,76  => 28 mod 48
  if 12 mod 16 = 1 mod 3: 28,76  => 28 mod 48
  if 1 mod 4 = 7 mod 9: 25,61,97 => 25 mod 36
  if 3 mod 4 = 7 mod 9: 7,43,79  =>  7 mod 36
triple steps:
  if 1 mod 12 = 25 mod 36        => 25 mod 36
  if 7 mod 12 =  7 mod 36        =>  7 mod 36