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import FormalConjecturesUtilA108306: 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:
namespace 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 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 6 := ⊢ a 1 = 6 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 21 := ⊢ a 2 = 21 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 81 := ⊢ a 3 = 81 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 306 := ⊢ a 4 = 306 All goals completed! 🐙open MatrixThe 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 => 2The 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 := n:ℕ⊢ a n = (m ^ (n + 1)) 0 0
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^kThe 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 => bThe 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 := a_val:ℕb_val:ℕn:ℕ⊢ invertSeqD a_val b_val n = (genMatrix a_val b_val ^ n) 0 0
All goals completed! 🐙end OeisA108306