Linear recurrence a(n) = 5a(n-1) - 8a(n-2) with a(0)=0, a(1)=1

Conjecture: Let S(p) := Σ_k=0^p-1 a(4k) C(2k, k)^3 / (-4096)^k. Then S(p) ≡ 0 pmodp^2 for every odd prime p. - _Zhi-Wei Sun_, Mar 13 2013

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-190969,
  title        = {Linear recurrence a(n) = 5a(n-1) - 8a(n-2) with a(0)=0, a(1)=1},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-190969}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. Conjecture: Let $S (p) := \sum_{k = 0}^{p - 1} a (4 k) (k 2 k)^{3} / (- 4096)^{k}$ . Then $S (p) \equiv 0 (mod p^{2})$ for every odd prime $p$ .

_Zhi-Wei Sun_, Mar 13 2013

conjecture_2. Conjecture: Let $S (p) := \sum_{k = 0}^{p - 1} a (4 k) (k 2 k)^{3} / (- 4096)^{k}$ . Then $S (p) \equiv 0 (mod p^{3})$ for any odd prime $p \equiv 1, 2, 4 (mod 7)$ .

_Zhi-Wei Sun_, Mar 13 2013

The sequence $a (n)$ satisfies the linear recurrence relation $a (n) = 5 a (n - 1) - 8 a (n - 2)$ with initial values $a (0) = 0$ and $a (1) = 1$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«190969» (2 statements).

theorem conjecture_1 (p : ℕ) (hp : p.Prime) (hp2 : p ≠ 2) :
    sMod p 2 = 0

theorem conjecture_2 (p : ℕ) (hp : p.Prime) (hp2 : p ≠ 2)
    (h7 : p % 7 ∈ ({1, 2, 4} : Finset ℕ)) :
    sMod p 3 = 0

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A190969
Conjectures involving primes and quadratic forms by Zhi-Wei Sun, arXiv:1211.1588 (2012)

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.