• the sum of F(i) to F(i+9) is equal to F(i+11) − F(i+1) — proof:

   F(i) + F(i+1) + F(i+2) + F(i+3) + F(i+4) + F(i+5) + F(i+6) + F(i+7) + F(i+8) + F(i+9)
   = F(i+2) + F(i+4) + F(i+6) + F(i+8) + F(i+10)
   = F(i+2) - F(i+3) + F(i+5) + F(i+6) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+7) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+9) + F(i+10)
   = -F(i+1) + F(i+11)
  

• F(i+246) mod 10 is equal to (F(i+11) − F(i+1)) mod 10 (discovered by experimentation; no idea how to prove this)

• the sum of F(i) to F(i+9) is equal to F(i+11) − F(i+1) — proof:

   F(i) + F(i+1) + F(i+2) + F(i+3) + F(i+4) + F(i+5) + F(i+6) + F(i+7) + F(i+8) + F(i+9)
   = F(i+2) + F(i+4) + F(i+6) + F(i+8) + F(i+10)
   = F(i+2) - F(i+3) + F(i+5) + F(i+6) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+7) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+9) + F(i+10)
   = -F(i+1) + F(i+11)
  

• F(i+246) mod 10 is equal to (F(i+11) − F(i+1)) mod 10 (discovered by experimentation; no idea how to prove this)

Perl, 78

sub F{$c=shift;$c>1?F($c-2)+F($c-1):$c}$_=<>;$x=F($_+11)-F($_+1);print$x+$x%10

This makes use of my observation that

• the sum of F(i) to F(i+9) is equal to F(i+11) − F(i+1) (please comment if you want a proof)

• F(i+246) mod 10 is equal to (F(i+11) − F(i+1)) mod 10 (discovered by experimentation; no idea how to prove this)

Perl, 78

sub F{$c=shift;$c>1?F($c-2)+F($c-1):$c}$_=<>;$x=F($_+11)-F($_+1);print$x+$x%10

This makes use of my observation that

• the sum of F(i) to F(i+9) is equal to F(i+11) − F(i+1) — proof:

   F(i) + F(i+1) + F(i+2) + F(i+3) + F(i+4) + F(i+5) + F(i+6) + F(i+7) + F(i+8) + F(i+9)
   = F(i+2) + F(i+4) + F(i+6) + F(i+8) + F(i+10)
   = F(i+2) - F(i+3) + F(i+5) + F(i+6) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+7) + F(i+8) + F(i+10)
   = -F(i+1) + F(i+9) + F(i+10)
   = -F(i+1) + F(i+11)
  

• F(i+246) mod 10 is equal to (F(i+11) − F(i+1)) mod 10 (discovered by experimentation; no idea how to prove this)