Pith's one-line read
Evenly spaced Fibonacci-type subsequences are claimed to have closed-form Frobenius numbers and genus.
desk verdict
The paper's even-d extension has a plausible Apéry-set framework, but the headline Frobenius formulas for Fibonacci and Lucas subsequences fail when n<d, and the counterexamples are decisive.
read the letter →
A machine-rendered reading of the paper's core claim, the
machinery that carries it, and where it could break.
The reading
This paper studies numerical semigroups generated by equally spaced terms of a generalized Fibonacci sequence: $S=\langle V_n, V_{n+d}, V_{n+2d},\ldots\rangle$, where $d$ is even and the usual gcd conditions hold. The aim is to determine the Frobenius number (the largest integer with no representation as a nonnegative combination of the generators) and the genus (the number of such nonrepresentable integers). The paper derives the embedding dimension, an explicit Apéry set via a greedy algorithm, and formulas for the Frobenius number in the general case. In the special cases $V_n=F_n$ (Fibonacci) and $V_n=L_n$ (Lucas), the Frobenius number is claimed to reduce to the closed forms $F(S_1)=F_dF_{2n}-F_{n+d}$ and $F(S_2)=F_d(L_{2n+1}+L_{2n-1})-L_{n+d}$, with a recurrence-based formula for the genus. These families matter because closed-form Frobenius data is rare for semigroups with more than two generators.
What carries the argument
The machinery has two parts. First, the greedy algorithm: for a target $x$, coefficients $\lambda_i$ are chosen from largest to smallest against $1,F_{2d}/F_d,\ldots,F_{kd}/F_d$, giving $s(x)=\sum_i\lambda_i V_{n+id}$, the least element of $S$ in the residue class $V_{n+d}x\pmod{V_n}$. Second, the Apéry-set identity $s(x)=V_{n+d}x-\lfloor F_{(k-1)d}x/F_{kd}\rfloor V_n$, which converts greedy data into an explicit numerical value and makes sums over Apéry elements tractable. The supporting Fibonacci identities in Proposition 2.1.2, especially $F_d V_{n+kd}-F_{kd}V_{n+d}=(-1)^{d-1}F_{(k-1)d}V_n$, carry the coefficient estimates. The embedding-dimension threshold $F_{\kappa d}/F_d\ge V_n$ fixes how many generators are minimal.
What would settle it
Compute $S_1$ for $n=3,d=8$: the generators are $F_3=2$ and $F_{11}=89$, so the two-generator formula gives $F(S_1)=2\cdot89-2-89=87$, whereas Corollary 3.4.2 gives $F_8F_6-F_{11}=21\cdot8-89=79$.
The central claim is that, for even $d$ and $\gcd(V_1,V_2)=\gcd(V_n,F_d)=1$, the semigroup $S=\langle V_n,V_{n+d},V_{n+2d},\ldots\rangle$ is completely described by the smallest $\kappa$ with $F_{\kappa d}/F_d\ge V_n$: the embedding dimension is $\kappa$, and the Apéry set with respect to $V_n$ is $\{s(x):1\le x\le V_n-1\}\cup\{0\}$, where $s(x)=V_{n+d}x-\lfloor F_{(k-1)d}x/F_{kd}\rfloor V_n$ comes from greedy coefficients against the normalized Fibonacci numbers $F_{id}/F_d$. From this, $F(S)=s(V_n-1)-V_n$. For Fibonacci generators, the paper obtains $F(S_1)=F_dF_{2n}-F_{n+d}$ for $n\ge3$; for Lucas generators, $F(S_2)=F_d(L_{2n+1}+L_{2n-1})-L_{n+d}$ for $n\ge4$. The genus is computed in the special case by solving joint first-order recurrences for the partial sums $A_k,B_k$ defined in Proposition 3.4.3.
Load-bearing premise
The closed-form formulas assume the same greedy coefficient pattern works when the starting Fibonacci index is smaller than the spacing between generators, a case the proof's quotient analysis does not cover.
Editorial extensions
If this is right
For the Fibonacci subsequence $S_1=\langle F_n,F_{n+d},\ldots\rangle$, the largest nonrepresentable integer is claimed to be $F_dF_{2n}-F_{n+d}$, so the answer is one arithmetic expression rather than a search.
For the Lucas subsequence $S_2=\langle L_n,L_{n+d},\ldots\rangle$, the analogous claim is $F_d(L_{2n+1}+L_{2n-1})-L_{n+d}$.
The embedding dimension equals the smallest $\kappa$ with $F_{\kappa d}/F_d\ge V_n$, so the number of minimal generators can be read off a Fibonacci quotient.
Because the Apéry set is explicit, the genus can be assembled from the partial sums $A_k,B_k$, which obey the first-order recurrences in Proposition 3.4.3.
The results extend the earlier $d=2$ and odd-$d$ cases to every even spacing, covering the direction left open by the predecessor study.
Reading between the lines
Editorial extensions of the paper, not claims the author makes directly.
The stated ranges $n\ge3$ and $n\ge4$ in Corollary 3.4.2 are broader than the range in which the coefficient-pattern proof runs; a test around $n<d$ would determine whether a missing hypothesis such as $n\ge d$ is needed.
If the intended hypothesis is indeed $n\ge d$, the same method should produce analogous explicit formulas for the genus, since the $A_k,B_k$ recurrences do not depend on the small-index boundary.
The embedding-dimension threshold in Corollary 3.1.2 has the same quotient-algebra structure, so its stated formulas likely need the same boundary repair; checking the semigroup $\langle F_3,F_{11}\rangle$ illustrates the issue.