Perfect Squares In a Recurrence Relationship

Question

If $$a_{n+3}=-a_{n+2}+2a_{n+1}+8a_n $$ for $$ a_0=a_1=a_2=1.$$ Then prove that $a_n$ is a perfect square.

Answer 1

trial

Thu Aug  2 21:26:39 PDT 2018
3   9 =  3^2
4   1 =   1 
5   25 =  5^2
6   49 =  7^2
7   9 =  3^2
8   289 =  17^2
9   121 =  11^2
10   529 =  23^2
11   2025 =  3^4 5^2
12   1 =   1 
13   8281 =  7^2 13^2
14   7921 =  89^2
15   8649 =  3^2 31^2
16   73441 =  271^2
17   7225 =  5^2 17^2
18   208849 =  457^2
19   393129 =  3^2 11^2 19^2
20   82369 =  7^2 41^2
21   2374681 =  23^2 67^2
22   935089 =  967^2
23   4473225 =  3^4 5^2 47^2
24   16394401 =  4049^2
25   32761 =  181^2
26   68541841 =  17^2 487^2
27   62678889 =  3^2 7^2 13^2 29^2
28   74666881 =  8641^2
29   599025625 =  5^4 11^2 89^2
30   51739249 =  7193^2
31   1743647049 =  3^2 31^2 449^2
32   3152036449 =  23^2 2441^2
33   749171641 =  101^2 271^2
34   19504077649 =  7^2 71^2 281^2
35   7210557225 =  3^6 5^2 17^2 37^2
36   37790971201 =  73^2 2663^2
37   132662764441 =  457^2 797^2
38   603635761 =  79^2 311^2
39   567049662729 =  3^2 11^2 19^2 1201^2
40   495459724321 =  409^2 1721^2
41   643468687225 =  5^2 7^2 13^2 41^2 43^2
42   4883848063249 =  257^2 8599^2
43   366767105769 =  3^2 23^2 67^2 131^2
44   14548678518529 =  17^2 89^2 2521^2
45   25255640199001 =  967^2 5197^2
46   6775853684209 =  2603047^2
47   160124854862025 =  3^4 5^2 31^2 47^2 193^2
48   55471974098401 =  7^4 97^2 1567^2
49   318984565099321 =  11^2 401^2 4049^2
50   1072958221993681 =  271^2 120871^2
Thu Aug  2 21:26:40 PDT 2018