Let d(n) be the Conway-type integer sequence. For K≥ -1, let I_K=n:2^K+1≤ n≤ 2^K+2, with initial domain I_-1=1,2. For each n, let n_1 and n_2 be the two parent spots occurring in the recurrence for d(n). Parent-spot generation conjecture. For every n∈ I_K with K≥ 1, both parent spots satisfy n_1,n_2∈ I_K-1. This describes the generation structure underlying the recurrence for d(n). The source reports that the claim has been numerically verified to high order of K, but gives no proof or further resolution.

The parent-spot generation conjecture for the Conway-type sequence d(n)

From papers

Let $d(n)$ be the Conway-type integer sequence. For $K\geq -1$ , let

I_K=\left\{n:2^{K+1}\leq n\leq 2^{K+2}\right\},

with initial domain $I_{-1}=\{1,2\}$ . For each $n$ , let $n_1$ and $n_2$ be the two parent spots occurring in the recurrence for $d(n)$ . Parent-spot generation conjecture. For every $n\in I_K$ with $K\geq 1$ , both parent spots satisfy

n_1,n_2\in I_{K-1}.

This describes the generation structure underlying the recurrence for $d(n)$ . The source reports that the claim has been numerically verified to high order of $K$ , but gives no proof or further resolution.

Progress summary

Open

No public discussion or published progress on this conjecture was found.

No public discussion or published progress was found, so the conjecture appears open with no recorded activity.

Current status (as of August 2026): The conjecture remains open, with no publicly recorded proof, counterexample, or verified progress.

Sources & referencesView supporting material

Primary source

Klaus Pinn, “Study of Meta-Fibonacci Integer Sequences by Continuous Self-Referential Functional Equations”, arXiv:2603.17509 (2026).

integer-sequences meta-fibonacci-sequences

1 week ago

Let

D(1)=D(2)=1,\qquad D(n)=D(D(n-1))+D(n-1-D(n-2)),

and set $d(n)=2D(n)-n$ . The two parent spots are

n_1=D(n-1)=\frac{n-1+d(n-1)}2,\qquad n_2=n-1-D(n-2)=\frac{n-d(n-2)}2.

The generation intervals in the conjecture are explicitly closed and overlapping:

I_K=\{n:2^{K+1}\le n\le2^{K+2}\}.

Direct iteration of the defining recurrence gives

(D(1),\ldots,D(16)) =(1,1,2,2,2,3,4,4,4,4,5,6,7,8,8,8).

Now take $K=3$ and $n=16$ . Then

n\in I_3=[16,32],\qquad I_2=[8,16],

but

n_1=D(15)=8\in I_2,\qquad n_2=15-D(14)=7\notin I_2.

Equivalently, $d(14)=2$ , and hence $n_2=(16-2)/2=7$ .

The same pair of parent spots $(8,7)$ appears in Table 1 of the source, which explicitly notes that boundary indices belong to two consecutive generation intervals. Therefore the asserted universal inclusion is false for the stated closed intervals. A version with half-open intervals or an exception at lower endpoints would be a different conjecture.

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Shivam Patel · 22 hours ago