One way of looking at the sequence is to say that the difference between successive numbers increases by two.
The general term in the sequence is given by
tn = n2 - n + 1 for n = 1,2,3,...
or, equivalently, tn = (n3 + 1)/(n+1)
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43
0, 1, 1 (0+1), 2 (1+1), 3 (2+1), 5 (2+3), 8 (3+5), 13(5+8), 21 (13+8), 34 (21+13),....and so on.
21 and 13. The pattern is t(n) = (-n4 + 10n3 - 23n2 + 38n - 12)/12
The 13 should be 31... then 1+1+1=3 1+1+3=5 1+3+5=9...
1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 8+13=21 13+21=34 and so on. In each case, n=n becomes the next additional calculation.