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[N. J. A. Sloane]

A141619  	 	 Problem of the week at school.  	 	
                        3, 2, 6, 3, 9, 5, 12, 7
	               AUTHOR 	Marina (kiev.marina(AT)gmail.com), Aug 23 2008
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The definition below (if you change index from 0 to 1) *fits* given 8
terms but probably your teacher meant something much simpler ;-)
a(1)=3;
for n>1
if n is even, then a(n) = prime(n/2)
else /* n is odd */ a(n) = 3 + a(n-2)
PARI could implement above as
j=[]; j=concat(j,3); for(n=2, 30, if((floor(n/2))==(ceil(n/2)),
j=concat(j,(prime(n/2))), j=concat(j,3+(j[n-2])))); j
  [3, 2, 6, 3, 9, 5, 12, 7, 15, 11, 18, 13, 21, 17, 24, 19, 27, 23,
30, 29, 33, 31, 36, 37, 39, 41, 42, 43, 45, 47]

Best, AP