[seqfan] I can confirm A005646[8]=1015.

[Robert Munafo]

I have worked with Andrew Weimholt to confirm his work on the
extension of A005646. This is one of the sequences from the 2nd book
(Encyclopedia of Integer Sequences), and had not been extended since
the late 1970's. The sequence starts 1,1,1,3,6,26,122, to which Andrew
added 1015, and I can now confirm that result.

The problem represented by this sequence grows very quickly. For N=8
one is checking all 7x8 matrices with values of 0 or 1 (in theory,
that is 2^56 or 7.2e16 combinations, although some optimizations can
be applied reducing this to about 1.2e13). With 16 threads my
workstation needs 35 minutes to calculate this answer, but Andrew's
implementation is faster.

In an earlier message Andrew listed the 26 solutions for N=6 and my
program generates the same:

  1  2  4  8 10   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2)|(0,1,3,4,5)
; (3)|(0,1,2,4,5) ; (4)|(0,1,2,3,5)
  1  2  4  8 11   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2)|(0,1,3,4,5)
; (3)|(0,1,2,4,5) ; (0,4)|(1,2,3,5)
  1  2  4  8 13   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2)|(0,1,3,4,5)
; (3)|(0,1,2,4,5) ; (0,1,4)|(2,3,5)
  1  2  4  9 12   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2)|(0,1,3,4,5)
; (0,3)|(1,2,4,5) ; (1,4)|(0,2,3,5)
  1  2  4  9 16   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2)|(0,1,3,4,5)
; (0,3)|(1,2,4,5) ; (1,2,4)|(0,3,5)
  1  2  5  a 15   (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (0,2)|(1,3,4,5)
; (1,3)|(0,2,4,5) ; (0,2,4)|(1,3,5)
  1  2  c 14      (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2,3)|(0,1,4,5)
; (2,4)|(0,1,3,5)
  1  2  c 15      (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (2,3)|(0,1,4,5)
; (0,2,4)|(1,3,5)
  1  2  d 15      (0)|(1,2,3,4,5) ; (1)|(0,2,3,4,5) ; (0,2,3)|(1,4,5)
; (0,2,4)|(1,3,5)
  1  3  c 14      (0)|(1,2,3,4,5) ; (0,1)|(2,3,4,5) ; (2,3)|(0,1,4,5)
; (2,4)|(0,1,3,5)
  1  6  a 12      (0)|(1,2,3,4,5) ; (1,2)|(0,3,4,5) ; (1,3)|(0,2,4,5)
; (1,4)|(0,2,3,5)
  1  6  a 13      (0)|(1,2,3,4,5) ; (1,2)|(0,3,4,5) ; (1,3)|(0,2,4,5)
; (0,1,4)|(2,3,5)
  1  6  a 14      (0)|(1,2,3,4,5) ; (1,2)|(0,3,4,5) ; (1,3)|(0,2,4,5)
; (2,4)|(0,1,3,5)
  1  6  a 15      (0)|(1,2,3,4,5) ; (1,2)|(0,3,4,5) ; (1,3)|(0,2,4,5)
; (0,2,4)|(1,3,5)
  1  6  b 16      (0)|(1,2,3,4,5) ; (1,2)|(0,3,4,5) ; (0,1,3)|(2,4,5)
; (1,2,4)|(0,3,5)
  1  7  b 13      (0)|(1,2,3,4,5) ; (0,1,2)|(3,4,5) ; (0,1,3)|(2,4,5)
; (0,1,4)|(2,3,5)
  3  5  9 11      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (0,3)|(1,2,4,5)
; (0,4)|(1,2,3,5)
  3  5  9 12      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (0,3)|(1,2,4,5)
; (1,4)|(0,2,3,5)
  3  5  9 13      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (0,3)|(1,2,4,5)
; (0,1,4)|(2,3,5)
  3  5  a 14      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (1,3)|(0,2,4,5)
; (2,4)|(0,1,3,5)
  3  5  a 15      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (1,3)|(0,2,4,5)
; (0,2,4)|(1,3,5)
  3  5  b 15      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (0,1,3)|(2,4,5)
; (0,2,4)|(1,3,5)
  3  5  f 17      (0,1)|(2,3,4,5) ; (0,2)|(1,3,4,5) ; (0,1,2,3)|(4,5)
; (0,1,2,4)|(3,5)
  3  c 15         (0,1)|(2,3,4,5) ; (2,3)|(0,1,4,5) ; (0,2,4)|(1,3,5)
  3  d 15         (0,1)|(2,3,4,5) ; (0,2,3)|(1,4,5) ; (0,2,4)|(1,3,5)
  7  b 15         (0,1,2)|(3,4,5) ; (0,1,3)|(2,4,5) ; (0,2,4)|(1,3,5)

I also get the same values for the triangle:

Andrew Weimholt wrote:
> Here's the triangle I get...
>
> 1
> 0 1
> 0 0 1
> 0 0 1 2
> 0 0 0 3 3
> 0 0 0 3 17 6
> 0 0 0 1 36 74 11
> 0 0 0 1 60 573 358 23

-- 
  Robert Munafo  --  mrob.com