1. Introduction
For integer
, consider a set of positive integers
. Denote by
the number of nonnegative solutions
of the linear equation
for a positive integer
n. Recently, the concept of
p-numerical semigroups was introduced together with their symmetric characteristics [
1].
is often called the
denumerant. For a given nonnegative integer
p, let
be the set of all nonnegative integer
n’s such that
. For the set of nonnegative integers
, the set
is finite if and only if
. Then, there exists the largest element
in
, which is called the
p-Frobenius number, and each element is called the
gap. For the so-called
p-numerical semigroup , the cardinality
and the sum
of
are called the
p-Sylvester number (or the
p-genus) and the
p-Sylvester sum, respectively. When
,
is the original numerical semigroup, and the 0-Frobenius number
is the original Frobenius number
. Finding the Frobenius and related values is the well-known linear Diophantine problem, posed by Sylvester [
2] but known as the Frobenius problem, is the problem to determine the Frobenius number
. The Frobenius problem has been also known as the
coin exchange problem (or postage stamp problem/chicken McNugget problem), which has a long history and is one of the problems that has attracted many people as well as experts. The genus
is often fundamental in the study of algebraic curves and commutative algebra.
For two variables
, it is shown that [
2,
3]
An explicit expression of the Sylvester sum
is given by Brown and Shiue [
4] for two variables
as
This result is extended in [
5] for the power sum of the set of gaps
, defined by
in the case of
. However, for three or more variables, it is very complicated to find a general explicit formula for the Frobenius number, Sylvester number, and Sylvester sum. Only for some special cases have explicit formulas been found, including arithmetic, geometric-like, Fibonacci, Mersenne, and triangular (see, e.g., [
6] and references therein). The study of semigroups of natural numbers generated by three elements and its applications to algebraic geometry can be seen in [
7]. Some inexplicit formulas for the Frobenius number in three variables can be seen in [
8].
When
, the situation becomes even more difficult. For two variables, it is still easy to find explicit formulas of
,
, and
. However, for three or more variables, no explicit formula had been found, but finally, in 2022, we succeeded in obtaining closed formulas for some special cases, including the triplets of triangular numbers [
9], repunits [
10], Fibonacci [
11], and Jacobsthal numbers [
12,
13].
We are interested in finding any explicit closed formula for
. In this paper, we give explicit formulas for the triples forming arithmetic progressions
, where
a and
d are positive integers with
. The main result is given as follows (Theorem 1). For
,
We also show their explicit closed formulas for the power sum
(Theorem 2) and the weighted sum
, defined by (Theorem 3)
By exploiting the theory developed in this paper, it may be possible to find values in
p-numerical semigroups for other special triplets and sequences. Several new advanced results when
have been achieved, e.g., in [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. Some applications to Pell sequences can be found in [
26,
27].
3. The Main Result
In this section, we shall show the main result and give its proof.
Theorem 1. Let a and d be integers with , , and . Then, for , Remark 2. When , the formulas reduce toandrespectively, which are [32] ((3.9) and (3.10)) when . Let for nonnegative integers and . In the following tables, this is denoted by for simplicity.
When
a is odd, as seen in [
32] (3.6),
(
) as the complete residue system (minimal system) modulo
a is given by
Table 1.
Concerning the complete residue system
, each congruent value modulo
a moves up one line to the upper right block. However, only the two values in the top row move to fill the gap below the first block (see
Table 2). Namely, for
and
Since
we can know that each element in
has exactly two expressions in terms of
. Note that each element minus
a has only one expression which is yielded from the right-hand side, that is,
Considering the maximal value, it is clear that
Concerning the complete residue system
, each congruent value modulo
a moves up one line to the upper right block (the third block). However, only the two values in the top row in the second block move to fill the gap below the first block (see
Table 3). Namely, for
and
Since
we can know that each element in
has exactly three expressions in terms of
. Note that each element minus
a has two expressions which are yielded from the right-hand side, that is,
Considering the maximal value, it is clear that
If this process is continued for
, when
, the state shown in
Table 4 is reached. Here, the shaded cell parts show the elements of
. Elements with the same residues modulo
a move in the following positions according to
.
Indeed, each element
in
has exactly
expressions because
But any element has expressions.
Considering the maximal value, it is clear that
However, such a process cannot be continued further than . When , the same residue of modulo a comes to the position and no element comes to the bottom left position . Thus, after that, the pattern shifts and is complicated, so it becomes difficult to determine where the maximum element of is.
Therefore, when
a is odd, for
,
Concerning the number of representations, we need the summation of the elements in
. The elements in the staircase are
and the elements in the large block are
When
a is even,
(
) is given by
Table 5.
Each congruent value modulo
a moves up one line to the upper right block. However, only the two values in the top row move to fill the gap below the first block. Namely, for
and
When the process is continued for
, the state shown in
Table 6 is reached. Here, the shaded cell parts show the elements of
. Elements with the same residues modulo
a move in the following positions according to
.
Each element
in
has exactly
expressions because
But any element has expressions.
Since the maximal value in
is at the position
, when
a is even, for
,
However, when , no element comes to the position or because there is no element of in the top row. Hence, after , the pattern is shifted and the situation becomes irregular and complicated.
Concerning the number of representations, the elements in the staircase are
and the elements in the large block are
In [
32] (3.10),
where integers
q and
r are determined as
4. Power Sums
More generally, we can show a formula for the
p-Sylvester power sum
so that
and
. Once we know the exact structure of every element in
, by applying Proposition 1, we can obtain the formula. Namely, we need to calculate
for
. From the previous section, when
n is odd,
Hence, by Proposition 1, we obtain the generalized Sylvester power sum for .
Theorem 2. Let a, d, p, and μ be integers with , , , , and . Then, when a is odd, we have In particular, when , Theorem 2 reduces the formula of the p-Sylvester sum of the triple forming an arithmetic progression.
Corollary 1. Let a and d be integers with , and . Then, for , when a is odd, we have 5. Weighted Sums
In this section, we consider the weighted sums whose numbers of representations are less than or equal to
p [
34,
35]:
where
and
is a positive integer.
Here, Eulerian numbers
appear in the generating function
with
and
, and have an explicit formula
Then, an explicit formula of the
p-weighted sum is given in terms of the elements in
[
29] (see also [
30]).
Lemma 2. Assume that and . Then, for a positive integer μ, In order to obtain the formula for the
p-Sylvester weighted power sum, we need to calculate
for
,
and
. We use the formula
where
denote the Stirling numbers of the second kind, calculated as
From the previous section, when
n is odd,
Hence, by Lemma 2, we obtain the generalized Sylvester weighted power sum for .
Theorem 3. Let a, d, p, λ, and μ be integers with , , , , , , and . Then, when a is odd, we have Remark 3. The case is not included in Theorem 3, but in Theorem 2.
7. Final Comments
It should not be thought that a similar repetitive process to [KLP,KP,KY] is simply going on in any triple. For example, it is known that some triples of Pell sequences do not follow a similar process but the formation of the elements of 0-Apéry set is different (in preparation).
In addition, it is still very difficult to find any explicit formula for four or more variables in the sequence of arithmetic progressions too. For the moment, only in the case of repunits [
10], for
explicit formulas about four and five repunits are obtained, though the structures are even more complicated than for three variables. One of the reasons for the difficulties lies in the following. In the case of three variables, for any
j (
),
. However, in the case of four and more variables, for some
j’s,
. This means that some elements in
and in
are overlapped.
Selmer [
32] found a formula of the Frobenius number for almost arithmetic sequences by generalizing the previous result (Roberts [
36] for
; Brauer [
37] for
). For a positive integer
h,
Selmer also gave an explicit formula for the Sylvester number
. Some formulas for the Sylvester sum
and its variations are given in [
38]. However, it is known that even when
(the sequence of consecutive odd numbers), we have not found any explicit form of
for general
.
Another problem is whether we can find any convenient formula when .