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COMMENT
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Next term has 306322 digits. For a(1)=1, a(2)=2, and a(3)=1,2,3, sequence ends with 1's: 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,... 1, 2, 2, 0, 0, 1, 1, 1, 1, 1, 1, 1,... 1, 2, 3, 3, 0, 0, 1, 1, 1, 1, 1, 1,...
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MATHEMATICA
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a=1; b=2; c=4; s={a, b, c}; Do[d=Binomial[Binomial[c, b], Binomial[b, a]]; AppendTo[s, d]; a=b; b=c; c=d, {3}]; s IntegerDigits[s[[ -1]]]//Length 306322
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