aₙ₊₃ = 2aₙ₊₂ - 2aₙ₊₁ + aₙ

aₙ₊₄
= 2aₙ₊₃ - 2aₙ₊₂ + aₙ₊₁
= 2(2aₙ₊₂ - 2aₙ₊₁ + aₙ) - 2aₙ₊₂ + aₙ₊₁
= 2aₙ₊₂ - 3aₙ₊₁ + 2aₙ

aₙ₊₅
= 2aₙ₊₄ - 2ₙ₊₃ + aₙ₊₂
= 2(2aₙ₊₂ - 3aₙ₊₁ + 2aₙ) - 2(2aₙ₊₂ - 2aₙ₊₁ + aₙ) + aₙ₊₂
= aₙ₊₂ - 2aₙ₊₁ + 2aₙ

aₙ₊₆
= 2aₙ₊₅ - 2aₙ₊₄ + aₙ₊₃
= 2(aₙ₊₂ - 2aₙ₊₁ + 2aₙ) - 2(2aₙ₊₂ - 3aₙ₊₁ + 2aₙ) + 2aₙ₊₂ - 2aₙ₊₁ + aₙ
= aₙ

This tells you that a term is equal to the one that is 6 terms after it.

i.e

aₙ₊₆ = aₙ

So, if you keep going back by 6 terms or multiples of 6 terms, you will always get the same result.

a2016 = a2010 = a2004 = a1998 = ...

Basically,

2016 ÷ 6 = 336

So,

a2016
= a(336 × 6)
= a(335 × 6)
= a(334×6)
= a6


Or

a2016
= a(6 + 335×6)
= a6


= a3 - 2a2 + 2a1
= 7777 - 2(222) + 2(11)
= 7777 - 444 + 22
= 7333 + 22
= 7355