slug: oeis-a329369-printed-row-sum-range bibkey: oeis2024a329369 doi: null url: https://oeis.org/A329369 triage: theorem motivation_gids:

• D5/S1/Recurrence/Parity/DyadicPowerRowClosedForm

The printed index range of the sixth A329369 sum

Problem

Write a(n) for A329369 and T(n,k) for the A373183 row coefficients. Mikhail Kurkov printed on June 05 2024 the assertion that

a(2^mn + q) = Sum over i from A001511(n+1) to A000120(n)+1 of T(n, i) * a(2^m(2^(i-1)-1) + q)

holds for n at least zero, m at least zero, and q at least zero. The same assertion appears in A373183 as its second conjecture. The question is whether the printed range for q is correct.

Motivation

The two entries define a(n) by a(2n+1)=a(n) and, for positive n, a(2n)=a(n)+a(n-2^f(n))+a(2n-2^f(n)) with f the two-adic valuation, and they define T through the row polynomials R(2n+1,x)=xR(n,x), R(2n,x)=x(R(n,x+1)-R(n,x)), R(0,x)=x. Both definitions are already carried in this repository, so the printed sum can be evaluated against them directly.

Gap

The sum is a decomposition of the index into a multiple of 2^m and a remainder q. Such a decomposition normally carries the restriction that q is below 2^m, and the printed text carries no such restriction. Nothing in the entry says which reading is intended.

Route

The rows at a positive power of two have the closed form R(2^k, x) = (2^k - 1)x + 2^kx^2, proved by induction: the difference operator sends the coefficient pair (a, c) to (a + c, 2c), and the row at index two is x + 2x^2. Reading off coefficients gives T(2^k,1) = 2^k - 1 and T(2^k,2) = 2^k, while A001511(2^k+1) = 1 and A000120(2^k) + 1 = 2 pin the summation range to the two indices one and two. Evaluating at m equal to zero and q equal to one makes the left side a(2^k+1) = 2^k - 1 and the right side (2^k - 1) + 3*2^k = 2^(k+2) - 1. These are never equal.

Falsifier

An index triple with q at least 2^m at which the two sides agree would not disturb the theorem, which asserts only that specific triples differ. A proof that the two sides agree for every q at least zero would contradict it.

Evidence

• Module: D5/S1/Recurrence/Parity/DyadicPowerRowClosedForm.lean.
• Closed form: R_two_pow.
• Counterexample family: printed_recurrence_ne_at_two_pow.
• Closed negation of the printed reading: not_kurkovRowRecurrence.
• The defining recurrences are derived, not assumed: b_zero, b_odd_index, b_even_index, the last from the frozen row identity in D5/S1/Digit/DyadicRowPolynomialRecurrence.

Triage

theorem. Only the printed reading with q unrestricted is settled, and it is settled in the negative. The reading in which q is below 2^m is a different assertion; it is not settled here and stays open. An arithmetic sweep over indices below 2^18, with n below 256 and m at most ten, found 2781 differing triples for the printed reading and every one of them had q at least 2^m, while the restricted reading produced none. That sweep is evidence about a bounded window, not a proof.

ASSUMED-UNVERIFIED

Whether the unrestricted range is an oversight rather than the intended assertion is not established; only the printed text is cited. No search for a published correction or for prior notice of this range was made, so priority is not claimed.