FormalConjectures.OEIS.«108306»

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module

public import FormalConjecturesUtil

A108306: Expansion of $(3x+1)/(1-3x-3x^2)$

This sequence satisfies the linear recurrence relation $a(0)=1$, $a(1)=6$, and $a(n) = 3a(n-1) + 3a(n-2)$ for $n \ge 2$.

References:

A108306

@[expose] public sectionnamespace OeisA108306

The primary defining sequence a. a n is the $n$-th term of the expansion of $(3x+1)/(1-3x-3x^2)$, satisfying $a(0)=1, a(1)=6, a(n)=3a(n-1)+3a(n-2)$.

def a : ℕ → ℕ
  | 0 => 1
  | 1 => 6
  | k + 2 => 3 * a (k + 1) + 3 * a k

@[category test, AMS 11]
theorem a_0 : a 0 = 1 := by⊢ a 0 = 1 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_1 : a 1 = 6 := by⊢ a 1 = 6 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_2 : a 2 = 21 := by⊢ a 2 = 21 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_3 : a 3 = 81 := by⊢ a 3 = 81 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_4 : a 4 = 306 := by⊢ a 4 = 306 decideAll goals completed! 🐙

open Matrix

The matrix for the specific case $a=5, b=2$. $$M = \begin{pmatrix} 1 & 5 \cr 1 & 2 \end{pmatrix}$$

def m : Matrix (Fin 2) (Fin 2) ℕ :=
  fun i j => match i, j with
  | 0, 0 => 1
  | 0, 1 => 5
  | 1, 0 => 1
  | 1, 1 => 2

The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)]. These results are a case (a=5, b=2) of the general conjecture below.


@[category textbook, AMS 11]
theorem a_is_invert_transform_case (n : ℕ) :
    a n = (m ^ (n + 1)) 0 0 := byn:ℕ⊢ a n = (m ^ (n + 1)) 0 0
  -- `m` satisfies `m ^ 2 = 3 m + 3`, so its powers satisfy the recurrence of `a`.
  have hm : ∀ n, m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n := fun n => byn✝:ℕn:ℕ⊢ m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0
    rw [pow_add,n✝:ℕn:ℕ⊢ m ^ n * m ^ 2 = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0 show m ^ 2 = 3 • m + 3 • (1 : Matrix (Fin 2) (Fin 2) ℕ) byn✝:ℕn:ℕ⊢ m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0 decideAll goals completed! 🐙n:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0, mul_add,n✝:ℕn:ℕ⊢ m ^ n * 3 • m + m ^ n * 3 • 1 = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0
      mul_smul_comm,n✝:ℕn:ℕ⊢ 3 • (m ^ n * m) + m ^ n * 3 • 1 = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0 mul_smul_comm,n✝:ℕn:ℕ⊢ 3 • (m ^ n * m) + 3 • (m ^ n * 1) = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0 mul_one,n✝:ℕn:ℕ⊢ 3 • (m ^ n * m) + 3 • m ^ n = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0 ← pow_succn✝:ℕn:ℕ⊢ 3 • m ^ (n + 1) + 3 • m ^ n = 3 • m ^ (n + 1) + 3 • m ^ nn:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0]n:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0n:ℕhm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a n = (m ^ (n + 1)) 0 0
  induction n using Nat.twoStepInduction with
  | zero =>zerohm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a 0 = (m ^ (0 + 1)) 0 0 decideAll goals completed! 🐙
  | one =>onehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ n⊢ a 1 = (m ^ (1 + 1)) 0 0 decideAll goals completed! 🐙
  | more k ih1 ih2 =>morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ a (k + 2) = (m ^ (k + 2 + 1)) 0 0
    rw [a,morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * a (k + 1) + 3 * a k = (m ^ (k + 2 + 1)) 0 0morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0 ih1,morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * a (k + 1) + 3 * (m ^ (k + 1)) 0 0 = (m ^ (k + 2 + 1)) 0 0morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0 ih2,morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (m ^ (k + 2 + 1)) 0 0morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0 show k + 2 + 1 = k + 1 + 2 byn:ℕ⊢ a n = (m ^ (n + 1)) 0 0morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0 omegaAll goals completed! 🐙morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0, hm (k + 1)morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0]morehm:∀ (n : ℕ), m ^ (n + 2) = 3 • m ^ (n + 1) + 3 • m ^ nk:ℕih1:a k = (m ^ (k + 1)) 0 0ih2:a (k + 1) = (m ^ (k + 1 + 1)) 0 0⊢ 3 * (m ^ (k + 1 + 1)) 0 0 + 3 * (m ^ (k + 1)) 0 0 = (3 • m ^ (k + 1 + 1) + 3 • m ^ (k + 1)) 0 0
    simp only [Matrix.add_apply, Matrix.smul_apply, smul_eq_mul]All goals completed! 🐙

The c sequence for the general INVERT transform conjecture. $c(1) = 1$, $c(k)=ab^(k-2)$ for $k \ge 2$.

def invertSeqC (a b : ℕ) : ℕ → ℕ
  | 0 => 0
  | 1 => 1
  | (k + 2) => a * b^k

The INVERT transform of the sequence c.

noncomputable def invertSeqD (a b : ℕ) (n : ℕ) : ℕ :=
  Nat.strongRecOn n
    (fun m ih =>
      if m = 0 then 1
      else ∑ i ∈ Finset.range m,
        if h : i < m then invertSeqC a b (m - i) * ih i h else 0)

The general 2x2 matrix [(1,a); (1,b)].

def genMatrix (a b : ℕ) : Matrix (Fin 2) (Fin 2) ℕ :=
  fun i j => match i, j with
  | 0, 0 => 1
  | 0, 1 => a
  | 1, 0 => 1
  | 1, 1 => b

The conjecture: The INVERT transform of a sequence starting $(1, a, ab, ab^2, ab^3, \ldots)$ is equivalent to extracting the upper left terms of powers of the 2x2 matrix [(1,a); (1,b)].

@[category research solved, AMS 11,
  formal_proof using lean4 at
    "https://github.com/KitaKen1/oeis-108306/blob/2e01e81b20a56880993e174ee4af0f0d7af37bdd/lean/OeisA108306FC.lean#L101-L106"]
theorem conjecture (a_val b_val n : ℕ) :
    invertSeqD a_val b_val n = (genMatrix a_val b_val ^ n) 0 0 := bya_val:ℕb_val:ℕn:ℕ⊢ invertSeqD a_val b_val n = (genMatrix a_val b_val ^ n) 0 0
  sorryAll goals completed! 🐙

end OeisA108306