Representations by squares and triangular numbers

DefinitionThe square theta series#✓fully proved

The square theta series is the formal power series \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\) in \(\mathbb{Z}[[q]]\): for each \(n \ge 0\), its coefficient of \(q^n\) is the number of integers \(m\) with \(m^2 = n\).

public noncomputable def squareThetaSeries : PowerSeries ℤ :=
  PowerSeries.mk fun n => (((Finset.Icc (-(n : ℤ)) n).filter fun m : ℤ => m ^ 2 = (n : ℤ)).card : ℤ)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Used by 4

Used byFirst coefficients of the square theta series · The square theta series is a unit · Representation numbers for sums of squares · The one-square counts are the coefficients of the square theta series

DefinitionTriangular theta series#✓fully proved

The triangular theta series is the formal power series \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\) with integer coefficients: for each natural number \(n\), its coefficient of \(q^n\) is the number of natural numbers \(m\) with \(m(m+1)/2 = n\).

public noncomputable def triangularThetaSeries : PowerSeries ℤ :=
  PowerSeries.mk fun n => (((Finset.range (n + 1)).filter fun m => m * (m + 1) / 2 = n).card : ℤ)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Used by 5

Used byEta-quotient form of the Gladun–Chaudhary theta coefficient · First coefficients of the triangular theta series · The triangular theta series is a unit · Representation numbers for sums of triangular numbers · Triangular representation numbers with one summand

TheoremFirst coefficients of the square theta series#✓fully proved

In the square theta series \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\), the coefficients of \(q^0\), \(q^1\), \(q^2\) and \(q^4\) are \(1\), \(2\), \(0\) and \(2\) respectively.

public theorem coeff_squareThetaSeries_small :
    PowerSeries.coeff 0 squareThetaSeries = 1 ∧ PowerSeries.coeff 1 squareThetaSeries = 2 ∧
      PowerSeries.coeff 2 squareThetaSeries = 0 ∧ PowerSeries.coeff 4 squareThetaSeries = 2

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 1 · Used by 1

Statement usesThe square theta series

Used byThe square theta series is a unit

TheoremFirst coefficients of the triangular theta series#✓fully proved

In the triangular theta series \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\), the coefficients of \(q^0\), \(q^1\), \(q^2\) and \(q^3\) are \(1\), \(1\), \(0\) and \(1\) respectively.

public theorem coeff_triangularThetaSeries_small :
    PowerSeries.coeff 0 triangularThetaSeries = 1 ∧ PowerSeries.coeff 1 triangularThetaSeries = 1 ∧
      PowerSeries.coeff 2 triangularThetaSeries = 0 ∧
        PowerSeries.coeff 3 triangularThetaSeries = 1

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 1 · Used by 1

Statement usesTriangular theta series

Used byThe triangular theta series is a unit

TheoremThe square theta series is a unit#✓fully proved

The square theta series \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\) is a unit of the ring \(\mathbb{Z}[[q]]\) of formal power series with integer coefficients, since its constant coefficient is \(1\).

public theorem isUnit_squareThetaSeries : IsUnit squareThetaSeries

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 2 · Used by 2

Statement usesThe square theta series

Proof usesFirst coefficients of the square theta series

Used byRepresentation numbers for sums of squares · The one-square counts are the coefficients of the square theta series

TheoremThe triangular theta series is a unit#✓fully proved

The triangular theta series \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\) is a unit of the ring \(\mathbb{Z}[[q]]\) of formal power series with integer coefficients, since its constant coefficient is \(1\).

public theorem isUnit_triangularThetaSeries : IsUnit triangularThetaSeries

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 2 · Used by 2

Statement usesTriangular theta series

Proof usesFirst coefficients of the triangular theta series

Used byRepresentation numbers for sums of triangular numbers · Triangular representation numbers with one summand

DefinitionRepresentation numbers for sums of squares#✓fully proved

For an integer \(k\) and a natural number \(n\), \(r_k(n)\) is the coefficient of \(q^n\) in \(\theta(q)^k\), where \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\). Since \(\theta\) has constant coefficient \(1\), it is a unit of \(\mathbb{Z}[[q]]\), so \(\theta^k\) is a power series for negative \(k\) as well.

For \(k \ge 1\), \(r_k(n)\) is the number of ways to write \(n\) as an ordered sum of \(k\) squares of integers.

public noncomputable def sumOfSquaresRepCount (k : ℤ) (n : ℕ) : ℤ :=
  PowerSeries.coeff n ((isUnit_squareThetaSeries.unit ^ k : (PowerSeries ℤ)ˣ) : PowerSeries ℤ)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 2 · Used by 8

Statement usesThe square theta series is a unit · The square theta series

Used byA prime between \(p/2\) and \(p\) divides the representation counts of \(p\) · Representability criterion from a window of counts · Divisibility of square representation counts by a large prime is periodic · Integer zeros of the sums-of-squares counting polynomial · The one-square counts are the coefficients of the square theta series · Window congruence for sums of squares · Six-squares moments of Dedekind zeta coefficients · The six-squares moment vanishes below \(1\)

DefinitionRepresentation numbers for sums of triangular numbers#✓fully proved

For an integer \(k\) and a natural number \(n\), \(t_k(n)\) is the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). Since \(\psi\) has constant coefficient \(1\), it is a unit of \(\mathbb{Z}[[q]]\), so \(\psi^k\) is a power series for negative \(k\) as well.

For \(k \ge 1\), \(t_k(n)\) is the number of ways to write \(n\) as an ordered sum of \(k\) triangular numbers \(m(m+1)/2\) with \(m \ge 0\).

public noncomputable def sumOfTriangularRepCount (k : ℤ) (n : ℕ) : ℤ :=
  PowerSeries.coeff n ((isUnit_triangularThetaSeries.unit ^ k : (PowerSeries ℤ)ˣ) : PowerSeries ℤ)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 2 · Used by 9

Statement usesThe triangular theta series is a unit · Triangular theta series

Used byA prime between \(p/2\) and \(p\) divides the representation counts of \(p\) · Representability criterion from a window of counts · Prime divisibility of triangular representation numbers persists under prime shifts · Composition formula for triangular representation numbers · Integer roots of triangular representation numbers · Window criterion for representability by triangular numbers · Triangular representation numbers with one summand · Recurrence for triangular representation numbers · Window congruence for sums of triangular numbers

TheoremThe one-square counts are the coefficients of the square theta series#✓fully proved

For an integer \(k\) and \(n \ge 0\), let \(r_k(n)\) be the coefficient of \(q^n\) in \(\theta(q)^k\), where \(\theta\) is the square theta series. Then for every \(n \ge 0\), \(r_1(n)\) equals the coefficient of \(q^n\) in \(\theta(q)\).

public theorem sumOfSquaresRepCount_one (n : ℕ) :
    sumOfSquaresRepCount 1 n = PowerSeries.coeff n squareThetaSeries

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 3

Statement usesThe square theta series · Representation numbers for sums of squares

Proof usesThe square theta series is a unit

TheoremTriangular representation numbers with one summand#✓fully proved

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). Then for every natural number \(n\), \(t_1(n)\) equals the coefficient of \(q^n\) in \(\psi(q)\).

public theorem sumOfTriangularRepCount_one (n : ℕ) :
    sumOfTriangularRepCount 1 n = PowerSeries.coeff n triangularThetaSeries

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Section 1.

Uses 3

Statement usesRepresentation numbers for sums of triangular numbers · Triangular theta series

Proof usesThe triangular theta series is a unit

TheoremDivisibility of square representation counts by a large prime is periodic#●ready to prove

For an integer \(k\) and \(n \ge 0\), let \(r_k(n)\) be the coefficient of \(q^n\) in \(\theta(q)^k\), where \(\theta\) is the square theta series; for \(k \ge 1\) it is the number of representations of \(n\) as an ordered sum of \(k\) squares of integers. Let \(n \ge 1\) and \(k \ge 1\) be integers and let \(p\) be a prime with \(p \ge n + 1\). If \(p\) divides \(r_k(n)\), then \(p\) divides \(r_{k+jp}(n)\) for every integer \(j\).

public theorem_wanted sumOfSquaresRepCount_add_mul_prime {n k p : ℕ} (hn : 1 ≤ n) (hk : 1 ≤ k)
    (hp : p.Prime) (hpn : n + 1 ≤ p) (h : (p : ℤ) ∣ sumOfSquaresRepCount k n) (r : ℤ) :
    (p : ℤ) ∣ sumOfSquaresRepCount (k + r * p) n

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 3.

Uses 1

Statement usesRepresentation numbers for sums of squares

TheoremPrime divisibility of triangular representation numbers persists under prime shifts#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). Let \(n \ge 1\) and \(k \ge 1\) be natural numbers and let \(p\) be a prime with \(p \ge n + 1\). If \(p\) divides \(t_k(n)\), then \(p\) divides \(t_{k+rp}(n)\) for every integer \(r\).

public theorem_wanted sumOfTriangularRepCount_add_mul_prime {n k p : ℕ} (hn : 1 ≤ n)
    (hk : 1 ≤ k) (hp : p.Prime) (hpn : n + 1 ≤ p)
    (h : (p : ℤ) ∣ sumOfTriangularRepCount k n) (r : ℤ) :
    (p : ℤ) ∣ sumOfTriangularRepCount (k + r * p) n

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 4.

Uses 1

Statement usesRepresentation numbers for sums of triangular numbers

TheoremWindow congruence for sums of squares#●ready to prove

For an integer \(k\), let \(r_k(n)\) be the coefficient of \(q^n\) in \(\theta(q)^k\), where \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\). Let \(n \ge 1\), let \(p\) be an odd prime with \(p - 1 \ge n + 1\), and let \(s \ge p\) be an integer. Then \(p\) divides \(\sum_{i=0}^{p-1} r_{s-i}(n)\).

public theorem_wanted sum_sumOfSquaresRepCount_window {n p : ℕ} (hn : 1 ≤ n) (hp : p.Prime)
    (hp2 : p ≠ 2) (hpn : n + 1 ≤ p - 1) {s : ℤ} (hs : (p : ℤ) ≤ s) :
    (p : ℤ) ∣ ∑ i ∈ Finset.range p, sumOfSquaresRepCount (s - i) n

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 5.

Uses 1

Statement usesRepresentation numbers for sums of squares

TheoremWindow congruence for sums of triangular numbers#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). Let \(n \ge 1\), let \(p\) be an odd prime with \(p - 1 \ge n + 1\), and let \(s \ge p\) be an integer. Then \(p\) divides \(\sum_{i=0}^{p-1} t_{s-i}(n)\).

public theorem_wanted sum_sumOfTriangularRepCount_window {n p : ℕ} (hn : 1 ≤ n)
    (hp : p.Prime) (hp2 : p ≠ 2) (hpn : n + 1 ≤ p - 1) {s : ℤ} (hs : (p : ℤ) ≤ s) :
    (p : ℤ) ∣ ∑ i ∈ Finset.range p, sumOfTriangularRepCount (s - i) n

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 6.

Uses 1

Statement usesRepresentation numbers for sums of triangular numbers

TheoremRepresentability criterion from a window of counts#●ready to prove

For an integer \(k\) and \(n \ge 0\), let \(r_k(n)\) and \(t_k(n)\) be the coefficients of \(q^n\) in \(\theta(q)^k\) and \(\psi(q)^k\), where \(\theta\) and \(\psi\) are the square and triangular theta series (units of \(\mathbb{Z}[[q]]\)). Let \(n \ge 1\), let \(p\) be an odd prime with \(p - 1 \ge n + 1\), and let \(s\) be an integer. If \(p\) does not divide \(r_{s-1}(n) + r_{s-2}(n) + \cdots + r_{s-(p-1)}(n)\), then \(r_s(n) \ne 0\). Likewise, if \(p\) does not divide \(t_{s-1}(n) + t_{s-2}(n) + \cdots + t_{s-(p-1)}(n)\), then \(t_s(n) \ne 0\).

The source places no condition on \(s\), so the statement is made for every integer \(s\).

public theorem_wanted repCount_ne_zero_of_window {n p : ℕ} (hn : 1 ≤ n) (hp : p.Prime)
    (hp2 : p ≠ 2) (hpn : n + 1 ≤ p - 1) (s : ℤ) :
    (¬ (p : ℤ) ∣ ∑ i ∈ Finset.Icc 1 (p - 1), sumOfSquaresRepCount (s - i) n →
      sumOfSquaresRepCount s n ≠ 0) ∧
    (¬ (p : ℤ) ∣ ∑ i ∈ Finset.Icc 1 (p - 1), sumOfTriangularRepCount (s - i) n →
      sumOfTriangularRepCount s n ≠ 0)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 7.

Uses 2

Statement usesRepresentation numbers for sums of squares · Representation numbers for sums of triangular numbers

TheoremWindow criterion for representability by triangular numbers#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). Let \(n \ge 1\) and let \(p\) be an odd prime with \(p - 1 \ge n + 1\). For every integer \(s\), if \(p\) does not divide \(\sum_{i=1}^{p-1} t_{s-i}(n)\), then \(t_s(n) \ne 0\).

public theorem_wanted sumOfTriangularRepCount_ne_zero_of_window {n p : ℕ} (hn : 1 ≤ n)
    (hp : p.Prime) (hp2 : p ≠ 2) (hpn : n + 1 ≤ p - 1) (s : ℤ)
    (h : ¬ (p : ℤ) ∣ ∑ i ∈ Finset.Icc 1 (p - 1), sumOfTriangularRepCount (s - i) n) :
    sumOfTriangularRepCount s n ≠ 0

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 8.

Uses 1

Statement usesRepresentation numbers for sums of triangular numbers

TheoremA prime between \(p/2\) and \(p\) divides the representation counts of \(p\)#●ready to prove

For integers \(k, n \ge 1\), let \(r_k(n)\) be the number of representations of \(n\) as an ordered sum of \(k\) squares of integers and \(t_k(n)\) the number of representations of \(n\) as an ordered sum of \(k\) triangular numbers \(m(m+1)/2\) with \(m \ge 0\). Let \(p\) be an odd prime and \(\ell\) a prime with \(p/2 < \ell < p\). Then \(\ell\) divides both \(r_\ell(p)\) and \(t_\ell(p)\).

public theorem_wanted prime_dvd_repCount_of_half_lt {p q : ℕ} (hp : p.Prime) (hp2 : p ≠ 2)
    (hq : q.Prime) (hpq : p < 2 * q) (hqp : q < p) :
    (q : ℤ) ∣ sumOfSquaresRepCount q p ∧ (q : ℤ) ∣ sumOfTriangularRepCount q p

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 11.

Uses 2

Statement usesRepresentation numbers for sums of squares · Representation numbers for sums of triangular numbers

DefinitionLogarithmic coefficients of the triangular theta series#✓fully proved

For a natural number \(n\), let \(b_n\) be the rational number \(b_n = \sum_{d \mid n,\ n/d \text{ odd}} 1/d - \sum_{d \mid n,\ n/d \text{ even}} 1/d\), where \(d\) runs over the positive divisors of \(n\); for \(n = 0\) the sums are taken to be empty, so \(b_0 = 0\).

With \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\), the expansion \(\log \psi(q) = \sum_{m \ge 1} \log(1 - q^{2m}) - \sum_{m \ge 1} \log(1 - q^{2m-1})\) in the proof of Theorem 2 shows that \(b_n\) is the coefficient of \(q^n\) in \(\log \psi(q)\). The printed formula of Theorem 2 has the parity condition on \(d\) instead of \(n/d\), which gives \(b_2 = 1/2\) instead of \(-1/2\).

public def triangularLogCoeff (n : ℕ) : ℚ :=
  ∑ d ∈ n.divisors, if Odd (n / d) then (1 : ℚ) / d else -((1 : ℚ) / d)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 2 and its proof.

Used by 3

Used byComposition formula for triangular representation numbers · Recurrence for triangular representation numbers · First logarithmic coefficients of the triangular theta series

TheoremFirst logarithmic coefficients of the triangular theta series#✓fully proved

The logarithmic coefficients \(b_n\) of the triangular theta series satisfy \(b_1 = 1\), \(b_2 = -1/2\) and \(b_3 = 4/3\); this matches the expansion \(\log(1 + q + q^3 + \cdots) = q - q^2/2 + 4q^3/3 + \cdots\).

public theorem triangularLogCoeff_small :
    triangularLogCoeff 1 = 1 ∧ triangularLogCoeff 2 = -1 / 2 ∧ triangularLogCoeff 3 = 4 / 3

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 2 and its proof.

Uses 1

Statement usesLogarithmic coefficients of the triangular theta series

TheoremComposition formula for triangular representation numbers#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). For \(m \ge 1\) let \(b_m = \sum_{d \mid m,\ m/d \text{ odd}} 1/d - \sum_{d \mid m,\ m/d \text{ even}} 1/d\), the coefficient of \(q^m\) in \(\log \psi(q)\). Then for every integer \(k\) and natural number \(n\),

\[t_k(n) = \sum_{(w_1, \dots, w_l)} \frac{k^l}{l!}\, b_{w_1} \cdots b_{w_l},\]

where the sum runs over all compositions \((w_1, \dots, w_l)\) of \(n\), that is, all tuples of positive integers with sum \(n\), of every length \(l\) (for \(n = 0\), only the empty composition).

The source prints \(b_m\) with the parity condition on \(d\) instead of \(m/d\); with that coefficient the formula fails, giving \(t_1(2) = 1\) although \(t_1(2) = 0\).

public theorem_wanted sumOfTriangularRepCount_eq_sum_compositions (k : ℤ) (n : ℕ) :
    (sumOfTriangularRepCount k n : ℚ) =
      ∑ c : Composition n, (k : ℚ) ^ c.length / (c.length.factorial : ℚ) *
        (c.blocks.map triangularLogCoeff).prod

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 2.

Uses 2

Statement usesRepresentation numbers for sums of triangular numbers · Logarithmic coefficients of the triangular theta series

TheoremRecurrence for triangular representation numbers#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). For \(m \ge 1\) let \(b_m = \sum_{d \mid m,\ m/d \text{ odd}} 1/d - \sum_{d \mid m,\ m/d \text{ even}} 1/d\), the coefficient of \(q^m\) in \(\log \psi(q)\). Then for every integer \(k\) and natural number \(n\),

\[t_k(n + 1) = \sum_{j=0}^{n} \frac{k (j + 1)\, b_{j+1}}{n + 1}\, t_k(n - j).\]

public theorem_wanted sumOfTriangularRepCount_succ (k : ℤ) (n : ℕ) :
    (sumOfTriangularRepCount k (n + 1) : ℚ) =
      ∑ j ∈ Finset.range (n + 1),
        (k : ℚ) * (j + 1) * triangularLogCoeff (j + 1) / (n + 1) * sumOfTriangularRepCount k (n - j)

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Corollary 1, second recurrence.

Uses 2

Statement usesRepresentation numbers for sums of triangular numbers · Logarithmic coefficients of the triangular theta series

TheoremInteger zeros of the sums-of-squares counting polynomial#●ready to prove

For an integer \(k\) and \(n \ge 0\), let \(r_k(n)\) be the coefficient of \(q^n\) in \(\theta(q)^k\), where \(\theta\) is the square theta series. For every \(n \ge 0\) there is a polynomial \(P(z) = \sum_i c_i z^i\) with integer coefficients such that \(P(z) = n! \, r_z(n)\) for every integer \(z\), and every nonzero integer \(k\) with \(r_k(n) = 0\) divides \(c_1\) and satisfies \(|k| < 2 |c_i|^{1/(n-i)}\) for some \(i\) with \(0 \le i < n\).

Equivalently, \(|k| < 2 R_n\) with \(R_n = \max_{0 \le i \le n-1} |c_i|^{1/(n-i)}\).

public theorem_wanted sumOfSquaresRepCount_integer_root (n : ℕ) :
    ∃ P : Polynomial ℤ, (∀ z : ℤ, P.eval z = n.factorial * sumOfSquaresRepCount z n) ∧
      ∀ k : ℤ, k ≠ 0 → sumOfSquaresRepCount k n = 0 →
        k ∣ P.coeff 1 ∧
          ∃ i < n, (|k| : ℝ) < 2 * (|P.coeff i| : ℝ) ^ ((1 : ℝ) / ((n : ℝ) - i))

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 9.

Uses 1

Statement usesRepresentation numbers for sums of squares

TheoremInteger roots of triangular representation numbers#●ready to prove

For an integer \(k\), let \(t_k(n)\) be the coefficient of \(q^n\) in \(\psi(q)^k\), where \(\psi(q) = \sum_{m \ge 0} q^{m(m+1)/2}\). For every natural number \(n\) there is a polynomial \(P(z) = \sum_i a_i z^i\) with integer coefficients such that \(P(z) = n!\, t_z(n)\) for every integer \(z\), and every nonzero integer \(k\) with \(t_k(n) = 0\) divides \(a_1\) and satisfies \(|k| < 2 |a_i|^{1/(n-i)}\) for some \(i < n\).

For \(n \ge 1\) the bound says \(|k| < 2 \max_{0 \le i \le n-1} |a_i|^{1/(n-i)}\).

public theorem_wanted sumOfTriangularRepCount_integer_root (n : ℕ) :
    ∃ P : Polynomial ℤ, (∀ z : ℤ, P.eval z = n.factorial * sumOfTriangularRepCount z n) ∧
      ∀ k : ℤ, k ≠ 0 → sumOfTriangularRepCount k n = 0 →
        k ∣ P.coeff 1 ∧
          ∃ i < n, (|k| : ℝ) < 2 * (|P.coeff i| : ℝ) ^ ((1 : ℝ) / ((n : ℝ) - i))

Sources

• Alegri and Christopher, On Formulas, Recurrences, and Congruences for Representations of Integers as Sums of Squares and Triangular Numbers, INTEGERS 26 (2026), Article A93, Theorem 10.

Uses 1

Statement usesRepresentation numbers for sums of triangular numbers

12 fully proved11 ready to prove

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