bibkey: hardin2011a185765
authors: R. H. Hardin
year: 2011
title: "OEIS A185765: six-column binary arrays with unequal neighboring subblock traces"
doi: null
url: https://oeis.org/A185765
claim: "Empirical: a(n)=a(n-1)+30*a(n-2)+24*a(n-3)-95*a(n-4)-15*a(n-5)+64*a(n-6)-116*a(n-7)+16*a(n-8)+16*a(n-9) for n>12"
strata_touched:
  - D5/S3/Combinatorics/TraceAvoidingArrays/HardinA185765Transfer
  - D5/S3/Combinatorics/TraceAvoidingArrays/HardinA185765Recurrence
license: citation-only
triage: anchor

OEIS A185765

Verified locator

Source: https://oeis.org/A185765

DOI: null. This is an OEIS entry with no associated arXiv identifier. The locators are the NAME and sole FORMULA fields in revision 9.

Source statement

Number of (n+1)X6 binary arrays with no 2X2 subblock trace equal to any horizontal or vertical neighbor 2X2 subblock trace.

Empirical: a(n)=a(n-1)+30a(n-2)+24a(n-3)-95a(n-4)-15a(n-5)+64a(n-6)-116a(n-7)+16a(n-8)+16a(n-9) for n>12

The offset is 1,1. Arrays have labeled rows and columns, binary entries, open rectangular boundaries, and no symmetry quotient. A block trace is the ordinary sum of its upper-left and lower-right entries in the natural numbers. Horizontal and vertical neighbors have block origins one unit apart. The recurrence embeds cardinalities in the integers before signed arithmetic.

Mathematical mechanism

Each successive pair of rows is a state. Horizontal comparisons restrict states; vertical comparisons restrict transitions between overlapping states. Summing continuation counts over all initial row pairs counts the literal arrays. For the polynomial

Q(X)=X^9-X^8-30X^7-24X^6+95X^5+15X^4-64X^3+116X^2-16X-16,

the continuation residual vanishes after five transitions. Its total also vanishes at exponents three and four. Linearity propagates the vector identity and gives the printed recurrence at every index at least thirteen.

Evidence boundary

Preregistration: https://github.com/the-omega-institute/trureturing/issues/22447. The finite experiment is in the-omega-institute/trureturing-experiments, commit 8c1ae78df7371e28241a930a79d387b013bbe604, docs/reports/six-column-trace-avoidance/check.py, SHA-256 4467b18b5567329731539b327f1fee36ca25bc52fb06fd9fa3b912acd0797a94. The recorded command python3 check.py exited 0. It found 680 active row pairs, 3456 transitions, agreement with all 200 supplied terms, and scalar residual -4160 at index twelve. These are computed results.

Minimality of the polynomial, other widths, and a formula uniform in the width remain open. Literature completeness is ASSUMED-UNVERIFIED: the bounded screen in the preregistration does not establish an exhaustive negative search or worldwide priority. The transfer method is classical.