MathDB

Suppose

(a_1,a_2,a_3,a_4)

is a 4-term sequence of real numbers satisfying the following two conditions:
[*]

a_3=a_2+a_1

and

a_4=a_3+a_2

; [*] there exist real numbers

a,b,c

such that

an^2+bn+c=\cos(a_n)

for all

n\in\{1,2,3,4\}

.
Compute the maximum possible value of

\cos(a_1)-\cos(a_4)

over all such sequences

(a_1,a_2,a_3,a_4)

Problem Statement