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NESTED RECURRENCE RELATIONS WITH CONOLLY-LIKE SOLUTIONS*

Combinatorics · math.CO · posed by Alejandro Erickson, Abraham Isgur, Bradley W. Jackson, Frank Ruskey, Stephen M. Tanny · open

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Statement

Conjecture 5.1. For β>0\beta>0 , the only 2-ary, order 2(α,β)2(\alpha,\beta) -Conolly recurrences are

Record

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  • NESTED RECURRENCE RELATIONS WITH CONOLLY-LIKE SOLUTIONS*
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Conjecture 5.1 in Erickson–Isgur–Jackson–Ruskey–Tanny asserts that, for β>0\beta>0, the complete list of 2-ary order-2 (α,β)(\alpha,\beta)-Conolly recurrences

    ⟨s;a,b:t;c,d⟩:R(n)=R(n−s−R(n−a)−R(n−b))+R(n−t−R(n−c)−R(n−d))\langle s;a,b:t;c,d\rangle:\quad R(n)=R(n-s-R(n-a)-R(n-b))+R(n-t-R(n-c)-R(n-d))

    is exactly the table printed in the paper for (α,β)=(−2,3),(0,2),(2,1)(\alpha,\beta)=(-2,3),(0,2),(2,1), up to the stated ordering conventions.

    Result: The conjecture is false. A missing (−2,3)(-2,3)-Conolly recurrence is

    ⟨0;7,9:13;20,22⟩\boxed{\langle 0;7,9:13;20,22\rangle}

    with the first 2222 terms of the (−2,3)(-2,3)-Conolly sequence as initial conditions:

    1,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,6,7,8,8,8.1,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,6,7,8,8,8.

    Let CC be the (−2,3)(-2,3)-Conolly sequence, so the value mm occurs

    −2+3(1+ν2(m))=1+3ν2(m)-2+3(1+\nu_2(m))=1+3\nu_2(m)

    times. If S(m)S(m) is the last position occupied by mm, then

    S(m)=∑j≤m(1+3ν2(j))=4m−3s2(m),S(m)=\sum_{j\le m}(1+3\nu_2(j))=4m-3s_2(m),

    where s2(m)s_2(m) is the binary digit sum.

    For n>22n>22, write m=C(n)m=C(n), n=S(m−1)+ρn=S(m-1)+\rho, and define

    p=4⌊m/8⌋+min⁡(m mod 8,4),q=m−p.p=4\big\lfloor m/8\big\rfloor+\min(m\bmod 8,4),\qquad q=m-p.

    A direct check from the displayed formula for SS gives

    C(n−C(n−7)−C(n−9))=p,C(n-C(n-7)-C(n-9))=p,

    and

    C(n−13−C(n−20)−C(n−22))=q.C(n-13-C(n-20)-C(n-22))=q.

    Hence

    C(n)=m=p+q=C(n−C(n−7)−C(n−9))+C(n−13−C(n−20)−C(n−22)).C(n)=m=p+q =C(n-C(n-7)-C(n-9))+C(n-13-C(n-20)-C(n-22)).

    Thus CC satisfies ⟨0;7,9:13;20,22⟩\langle0;7,9:13;20,22\rangle, and the recurrence is well-defined from the stated initial conditions.

    This recurrence is not in the conjectured (−2,3)(-2,3) list: its normalized parameters (0;7,9:13;20,22)(0;7,9:13;20,22) have t=13t=13, whereas the conjectured table only includes t=1,3,5t=1,3,5 cases with the displayed bounded parameter sets.

    Citation: Original conjecture and notation: A. Erickson, A. Isgur, B. W. Jackson, F. Ruskey, S. M. Tanny, “Nested Recurrence Relations With Conolly-Like Solutions,” SIAM J. Discrete Math. 26 (2012), 206–238; arXiv:1509.02613, Conjecture 5.1.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The proposed counterexample attacks the correct Conjecture 5.1. The (−2,3)(-2,3)-Conolly sequence has multiplicities 1+3ν2(m)1+3\nu_2(m) and endpoint formula S(m)=4m−3s2(m)S(m)=4m-3s_2(m). The stated argument shows that for all n>22n>22 the two recursive arguments land in the blocks for pp and qq, with p+q=C(n)p+q=C(n), so the recurrence satisfies the required sequence from the given initial conditions.

    The recurrence ⟨0;7,9:13;20,22⟩\langle0;7,9:13;20,22\rangle is not in the conjectured (−2,3)(-2,3) table, whose listed cases have t=1,3,5t=1,3,5. I found no prior matching or stronger published result in the accessible literature searches.

    Novelty assessment

    TYPE1

    Classification rationale: This appears genuinely new, but minor. It is a short counterexample to a niche empirical classification conjecture, not a full classification or a new general method. It might merit mention as an erratum or short note, but likely not a standalone standard combinatorics paper by itself.

    Literature check: I found no accessible source proving that ⟨0;7,9:13;20,22⟩\langle0;7,9:13;20,22\rangle is (−2,3)(-2,3)-Conolly, nor any prior statement that Conjecture 5.1 is false. Searches covered exact recurrence strings, “Conjecture 5.1” with “Conolly-like,” OpenAlex/citation records for the original paper, OEIS entries, and related meta-Fibonacci/nested-recursion literature.

    Closest related result: Isgur–Kuznetsov–Rahman–Tanny (2014) contains a broad family that implicitly includes the same formal recurrence parameters, but with different initial conditions and a different frequency sequence; it does not imply this (−2,3)(-2,3)-Conolly counterexample.

    Citation: Original conjecture: A. Erickson, A. Isgur, B. W. Jackson, F. Ruskey, S. M. Tanny, “Nested Recurrence Relations with Conolly-like Solutions,” SIAM J. Discrete Math. 26 (2012), 206–238, Conjecture 5.1.

    Closest related family: A. Isgur, V. Kuznetsov, M. Rahman, S. Tanny, “Nested Recursions, Simultaneous Parameters and Tree Superpositions,” Electron. J. Combin. 21(1) (2014), #P1.49.

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