MHB Sequence Challenge: Find $a_{61}+a_{63}$

[anemone]

A sequence is defined recursively by $a_1=2007$, $a_2=2008$, $a_3=-2009$, and for $n>3$, $a_n=a_{n-1}-a_{n-2}+a_{n-3}+n$.

Find $a_{61}+a_{63}$.

 

[kaliprasad]

A sequence is defined recursively by $a_1=2007$, $a_2=2008$, $a_3=-2009$, and for $n>3$, $a_n=a_{n-1}-a_{n-2}+a_{n-3}+n$.

Find $a_{61}+a_{63}$.

Spoiler

we have

$a_n+a_{n-2}=a_{n-1} + a_{n-3} + n$

hence
$a_{63}+a_{61}=a_{62} + a_{60} + 63$
= $a_{61} + a_{59} + 62 + 63 $
= $a_{3} + a_{1} +4 \cdots+ 62 + 63 $
= - 2009 + 2007 + 63 * 64/2 - 6= 63 * 32 - 8 = 2008