bibkey: ratajczak2021a110491 authors: Lechoslaw Ratajczak year: 2021 title: “OEIS A110491, T-transform determinant conjecture for A093178” doi: null url: https://oeis.org/A110491 claim: “Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j.” strata_touched:

• D5/S1/Recurrence/Algebraic/RatajczakTTransformDeterminant license: CC-BY-SA-4.0 triage: anchor

OEIS A110491 and its T-transform conjecture

Lechoslaw Ratajczak added the following COMMENT to A110491 on August 4, 2021:

Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j.

The source uses one-based matrix indices and positive orders. A093178 is the sequence b(r)=1 for even r and b(r)=r for odd r. Consequently the lower branch is printed as b(2*j) even though its values are one. The Lean definition retains that literal branch. A110491 is characterized separately by

a(0)=1, a(1)=2, and a(m+2)=2*a(m+1)+4*(m+1)*m*a(m) for every nonnegative m.

Thus the formal result at index m identifies the determinant of the source’s order-m+1 matrix with a(m). Neither the recurrence nor the matrix is defined in terms of the other.

Verified locator

• URL: https://oeis.org/A110491
• A110491 revision 51 was supplied with SHA-256 177954d0a7a3e5640772d1a5f87c6a8cb6a6a00a5af6b3cdb2792fda3493a144.
• A093178 revision 82 was supplied with SHA-256 28ffb50cb61c1631274bf09cc426f61bf3ef2d50b2e53f476a904cef0058ed89.
• The OEIS material is attributed under the OEIS End-User License, Creative Commons Attribution-ShareAlike 4.0.

The scalar recurrence and the OEIS formula are published source material; they are not new results of the formalization. The new result recorded by the repository is the all-order proof that the literal determinant satisfies that independently given recurrence and initial values.

Prior-work boundary

The supplied bounded audit read the 15-page April 19, 2004 author copy of Plouffe’s A search for a mathematical expression for mass ratios using a large database 2004, linked from the author’s Articles directory under the same title cited by OEIS A093178. The inspected copy discusses numerical mass-ratio expression searches, constant databases, PSLQ, and Fibonacci and Lucas approximations; it contains no inspected T-transform or A110491 determinant theorem. The copy was downloaded from https://plouffe.fr/articles/A%20search%20for%20a%20mathematical%20expression%20for%20mass%20ratios%20using%20a%20large%20database%202004.pdf as 321707 bytes with SHA-256 c2f53b56dc78c89cdcee068d7553939ed0eb170399df96f256fe162f3378586b. The official viXra 1409.0099v1 endpoint returned HTTP 200 with media type application/pdf but a zero-byte body, so that version was not read; exact byte identity with the inspected author copy is ASSUMED-UNVERIFIED. This is a bounded reading result, not an exhaustive literature-search, worldwide-priority, or originality claim.