Alternating sums of Motzkin numbers

TheoremLog-convexity of the even-indexed Riordan numbers#●ready to prove

Let \(r_n\) denote the \(n\)-th Riordan number. The even-indexed Riordan numbers \(r_0, r_2, r_4, \dots\) form a log-convex sequence: for every natural number \(n\),

\[r_{2n+2}^2 \le r_{2n} \, r_{2n+4}.\]

The Riordan numbers begin \(1, 0, 1, 1, 3, 6, 15, \dots\).

public theorem_wanted riordanNumber_even_logConvex (n : ℕ) :
    riordanNumber (2 * (n + 1)) ^ 2 ≤
      riordanNumber (2 * n) * riordanNumber (2 * (n + 2))
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Corollary 1.
Uses 1
Statement usesRiordan numbers

DefinitionAlternating sums of Motzkin numbers#✓fully proved

Let \(M_k\) denote the \(k\)-th Motzkin number. For a natural number \(n\), the \(n\)-th alternating Motzkin sum is the integer

\[T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k.\]

It is OEIS A187306, which begins \(1, 0, 2, 2, 7, 14, 37, \dots\)

public def motzkinAlternatingSum (n : ℕ) : ℤ :=
  ∑ k ∈ Finset.range (n + 1), (-1) ^ (n - k) * (motzkinNumber k : ℤ)
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Equation (2).
Uses 1 · Used by 6
Statement usesMotzkin numbers as a Catalan-weighted binomial sum
Used byThe alternating Motzkin sums are eventually log-balanced and tempered · Products of consecutive alternating Motzkin sums are eventually log-convex and log-balanced · Alternating Motzkin sums at 1 and 2 · Recurrence for alternating Motzkin sums · The alternating Motzkin sum at 0 · Log-balancedness of Motzkin numbers plus their alternating sums

TheoremThe alternating Motzkin sum at 0#✓fully proved

Let \(M_k\) denote the \(k\)-th Motzkin number and let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\). Then \(T_0 = 1\).

public theorem motzkinAlternatingSum_zero : motzkinAlternatingSum 0 = 1
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Table 2.
Uses 3 · Used by 1
Statement usesAlternating sums of Motzkin numbers
Proof usesMotzkin numbers as a Catalan-weighted binomial sum · Value of the Motzkin number at 0
Used byAlternating Motzkin sums at 1 and 2

TheoremRecurrence for alternating Motzkin sums#✓fully proved

Let \(M_k\) denote the \(k\)-th Motzkin number and let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\). For every natural number \(n\), \(T_{n+1} = M_{n+1} - T_n\).

public theorem motzkinAlternatingSum_succ (n : ℕ) :
    motzkinAlternatingSum (n + 1) = motzkinNumber (n + 1) - motzkinAlternatingSum n
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Equation (2).
Uses 2 · Used by 1
Statement usesAlternating sums of Motzkin numbers · Motzkin numbers as a Catalan-weighted binomial sum
Used byAlternating Motzkin sums at 1 and 2

TheoremAlternating Motzkin sums at 1 and 2#✓fully proved

Let \(M_k\) denote the \(k\)-th Motzkin number and let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\). Then \(T_1 = 0\) and \(T_2 = 2\).

public theorem motzkinAlternatingSum_one_two :
    motzkinAlternatingSum 1 = 0 ∧ motzkinAlternatingSum 2 = 2
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Table 2.
Uses 6
Statement usesAlternating sums of Motzkin numbers
Proof usesRecurrence for alternating Motzkin sums · The alternating Motzkin sum at 0 · Motzkin numbers as a Catalan-weighted binomial sum · Value of the Motzkin number at 1 · Value of the Motzkin number at 2

DefinitionAlternating sums of Fibonacci numbers#✓fully proved

Let \(F_k\) denote the \(k\)-th Fibonacci number, with \(F_0 = 0\) and \(F_1 = 1\). For a natural number \(n\), the \(n\)-th alternating Fibonacci sum is the integer

\[W_n = \sum_{k=0}^{n} (-1)^{n-k} F_k.\]

The sequence begins \(0, 1, 0, 2, 1, 4, 4, 9, \dots\)

public def fibonacciAlternatingSum (n : ℕ) : ℤ :=
  ∑ k ∈ Finset.range (n + 1), (-1) ^ (n - k) * (Nat.fib k : ℤ)
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Equation (9).
Used by 3
Used byRecurrence for alternating Fibonacci sums · The alternating Fibonacci sums are eventually tempered · The alternating Fibonacci sum at 0

TheoremThe alternating Fibonacci sum at 0#✓fully proved

Let \(F_k\) denote the \(k\)-th Fibonacci number, with \(F_0 = 0\) and \(F_1 = 1\), and let \(W_n = \sum_{k=0}^{n} (-1)^{n-k} F_k\). Then \(W_0 = 0\).

public theorem fibonacciAlternatingSum_zero : fibonacciAlternatingSum 0 = 0
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Table 3.
Uses 1
Statement usesAlternating sums of Fibonacci numbers

TheoremRecurrence for alternating Fibonacci sums#✓fully proved

Let \(F_k\) denote the \(k\)-th Fibonacci number, with \(F_0 = 0\) and \(F_1 = 1\), and let \(W_n = \sum_{k=0}^{n} (-1)^{n-k} F_k\). For every natural number \(n\), \(W_{n+1} = F_{n+1} - W_n\).

public theorem fibonacciAlternatingSum_succ (n : ℕ) :
    fibonacciAlternatingSum (n + 1) = Nat.fib (n + 1) - fibonacciAlternatingSum n
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Equation (9).
Uses 1
Statement usesAlternating sums of Fibonacci numbers

TheoremThe alternating Motzkin sums are eventually log-balanced and tempered#●ready to prove

Let \(M_k\) denote the \(k\)-th Motzkin number and let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\). For every natural number \(n \ge 9\),

\[T_n^2 \le T_{n-1} T_{n+1} \quad\text{and}\quad n T_{n-1} T_{n+1} \le (n+1) T_n^2,\]

and for every natural number \(n \ge 6\),

\[n T_n^2 \le (n+1) T_{n-1} T_{n+1} \quad\text{and}\quad n T_{n-1} T_{n+1} \le (n+1) T_n^2.\]

Following Došlić, a positive sequence \((z_n)_{n \ge N}\) is log-balanced if it is log-convex, \(z_n^2 \le z_{n-1} z_{n+1}\), and \((z_n/n!)\) is log-concave, \(n z_{n-1} z_{n+1} \le (n+1) z_n^2\), for every \(n \ge N + 1\); it is tempered if its quotients \(x_n = z_{n+1}/z_n\) satisfy \(\frac{n}{n+1} x_{n-1} \le x_n \le \frac{n+1}{n} x_{n-1}\) for every \(n \ge N + 1\). So the two parts say that \((T_n)_{n \ge 8}\) is log-balanced and \((T_n)_{n \ge 5}\) is tempered, with the inequalities cleared of denominators (all terms are positive here) and the factorials and quotient weights using the original index \(n\), as in the paper's proofs.

public theorem_wanted motzkinAlternatingSum_logBalanced_tempered :
    (∀ n, 9 ≤ n →
      motzkinAlternatingSum n ^ 2 ≤
          motzkinAlternatingSum (n - 1) * motzkinAlternatingSum (n + 1) ∧
        n * (motzkinAlternatingSum (n - 1) * motzkinAlternatingSum (n + 1)) ≤
          (n + 1) * motzkinAlternatingSum n ^ 2) ∧
    (∀ n, 6 ≤ n →
      n * motzkinAlternatingSum n ^ 2 ≤
          (n + 1) * (motzkinAlternatingSum (n - 1) * motzkinAlternatingSum (n + 1)) ∧
        n * (motzkinAlternatingSum (n - 1) * motzkinAlternatingSum (n + 1)) ≤
          (n + 1) * motzkinAlternatingSum n ^ 2)
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Theorem 1.
Uses 1
Statement usesAlternating sums of Motzkin numbers

TheoremProducts of consecutive alternating Motzkin sums are eventually log-convex and log-balanced#●ready to prove

Products of consecutive alternating Motzkin sums are eventually log-convex and log-balanced. Let \(M_k\) denote the \(k\)-th Motzkin number, let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\) and let \(P_n = T_n T_{n+1}\). For every natural number \(n \ge 7\), \(P_n^2 \le P_{n-1} P_{n+1}\); and for every natural number \(n \ge 10\), \(P_n^2 \le P_{n-1} P_{n+1}\) and \(n P_{n-1} P_{n+1} \le (n+1) P_n^2\).

Following Došlić, a positive sequence \((z_n)_{n \ge N}\) is log-convex if \(z_n^2 \le z_{n-1} z_{n+1}\) for every \(n \ge N + 1\), and log-balanced if moreover \((z_n/n!)\) is log-concave, \(n z_{n-1} z_{n+1} \le (n+1) z_n^2\), for every \(n \ge N + 1\). So the two parts say that \((P_n)_{n \ge 6}\) is log-convex and \((P_n)_{n \ge 9}\) is log-balanced.

public theorem_wanted motzkinAlternatingSum_mul_succ_logConvex_logBalanced :
    (∀ n, 7 ≤ n →
      (motzkinAlternatingSum n * motzkinAlternatingSum (n + 1)) ^ 2 ≤
        (motzkinAlternatingSum (n - 1) * motzkinAlternatingSum n) *
          (motzkinAlternatingSum (n + 1) * motzkinAlternatingSum (n + 2))) ∧
    (∀ n, 10 ≤ n →
      (motzkinAlternatingSum n * motzkinAlternatingSum (n + 1)) ^ 2 ≤
          (motzkinAlternatingSum (n - 1) * motzkinAlternatingSum n) *
            (motzkinAlternatingSum (n + 1) * motzkinAlternatingSum (n + 2)) ∧
        n * ((motzkinAlternatingSum (n - 1) * motzkinAlternatingSum n) *
            (motzkinAlternatingSum (n + 1) * motzkinAlternatingSum (n + 2))) ≤
          (n + 1) * (motzkinAlternatingSum n * motzkinAlternatingSum (n + 1)) ^ 2)
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Theorem 2.
Uses 1
Statement usesAlternating sums of Motzkin numbers

TheoremLog-balancedness of Motzkin numbers plus their alternating sums#●ready to prove

Let \(M_n\) denote the \(n\)-th Motzkin number, let \(T_n = \sum_{k=0}^{n} (-1)^{n-k} M_k\) be the alternating sum of Motzkin numbers, and put \(z_n = M_n + T_n\). Then the sequence \((z_n)_{n \ge 6}\) is log-balanced: for every natural number \(n \ge 7\),

\[z_n^2 \le z_{n-1} \, z_{n+1} \quad\text{and}\quad n \, z_{n-1} \, z_{n+1} \le (n+1) \, z_n^2.\]

The first inequality is log-convexity; for positive terms the second is log-concavity of \(z_n / n!\) with denominators cleared, the factorial using the original index \(n\).

public theorem_wanted motzkinNumber_add_motzkinAlternatingSum_logBalanced (n : ℕ) (hn : 7 ≤ n) :
    ((motzkinNumber n : ℤ) + motzkinAlternatingSum n) ^ 2 ≤
        ((motzkinNumber (n - 1) : ℤ) + motzkinAlternatingSum (n - 1)) *
          ((motzkinNumber (n + 1) : ℤ) + motzkinAlternatingSum (n + 1)) ∧
      n * (((motzkinNumber (n - 1) : ℤ) + motzkinAlternatingSum (n - 1)) *
          ((motzkinNumber (n + 1) : ℤ) + motzkinAlternatingSum (n + 1))) ≤
        (n + 1) * ((motzkinNumber n : ℤ) + motzkinAlternatingSum n) ^ 2
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Theorem 3.
Uses 2
Statement usesAlternating sums of Motzkin numbers · Motzkin numbers as a Catalan-weighted binomial sum

TheoremThe alternating Fibonacci sums are eventually tempered#●ready to prove

Let \(F_k\) denote the \(k\)-th Fibonacci number, with \(F_0 = 0\) and \(F_1 = 1\), and let \(W_n = \sum_{k=0}^{n} (-1)^{n-k} F_k\). For every natural number \(n \ge 12\),

\[n W_n^2 \le (n+1) W_{n-1} W_{n+1} \quad\text{and}\quad n W_{n-1} W_{n+1} \le (n+1) W_n^2.\]

These are the conditions \(\frac{n}{n+1} x_{n-1} \le x_n \le \frac{n+1}{n} x_{n-1}\) on the quotients \(x_n = W_{n+1}/W_n\), cleared of denominators (all terms are positive here), so the sequence \((W_n)_{n \ge 11}\) is tempered, with the conditions imposed at every index \(n \ge 12\).

public theorem_wanted fibonacciAlternatingSum_tempered (n : ℕ) (hn : 12 ≤ n) :
    n * fibonacciAlternatingSum n ^ 2 ≤
        (n + 1) * (fibonacciAlternatingSum (n - 1) * fibonacciAlternatingSum (n + 1)) ∧
      n * (fibonacciAlternatingSum (n - 1) * fibonacciAlternatingSum (n + 1)) ≤
        (n + 1) * fibonacciAlternatingSum n ^ 2
Sources
  • Feng-Zhen Zhao, The Log-Balancedness of the Sequence for the Alternating Sums of Motzkin Numbers, INTEGERS 26 (2026), Article A55, Theorem 4.
Uses 1
Statement usesAlternating sums of Fibonacci numbers
7 fully proved5 ready to prove

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