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import FormalConjecturesUtilSum of fourth powers of Fibonacci-like binomial coefficients
The sequence is defined by $$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4.$$
References:
namespace OeisA181546open Filter RealThe generalized sum $F(n, L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$.
def F (n L : ℕ) : ℕ :=
∑ k ∈ Finset.range (n / 2 + 1), ((n - k).choose k) ^ L$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4$.
def a (n : ℕ) : ℕ := F n 4Lucas numbers $L(0) = 2, L(1) = 1, L(n) = L(n-1) + L(n-2)$.
def lucas : ℕ → ℕ
| 0 => 2
| 1 => 1
| n + 2 => lucas (n + 1) + lucas nConjectured limit value for $F(n+1, L) / F(n, L)$.
noncomputable def limitValue (L : ℕ) : ℝ :=
(Nat.fib L * sqrt 5 + lucas L) / 2
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 17 := ⊢ a 3 = 17 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 83 := ⊢ a 4 = 83 All goals completed! 🐙Conjecture: Given $F(n,L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$, then $\lim_{n\to\infty} F(n+1,L)/F(n,L) = (\mathrm{Fibonacci}(L)\sqrt{5} + \mathrm{Lucas}(L))/2$ for $L \ge 0$ where $\mathrm{Fibonacci}(n) = \mathrm{A000045}(n)$ and $\mathrm{Lucas}(n) = \mathrm{A000032}(n)$.
@[category research open, AMS 11]
theorem conjecture (L : ℕ) :
Tendsto (fun n => (F (n + 1) L : ℝ) / (F n L : ℝ)) atTop (nhds (limitValue L)) := L:ℕ⊢ Tendsto (fun n ↦ ↑(F (n + 1) L) / ↑(F n L)) atTop (nhds (limitValue L))
All goals completed! 🐙end OeisA181546