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2006 SMT Team Round #2 - Second Term in Sequence

Source:

August 22, 2011

Problem Statement

In a given sequence {S1,S2,...,Sk}\{S_1,S_2,...,S_k\}, for terms n≥3n\ge3, Sn=∑i=1n−1i⋅Sn−iS_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}. For example, if the first two elements are 2 and 3, respectively, the third entry would be 1⋅3+2⋅2=71\cdot3+2\cdot2=7, and the fourth would be 1⋅7+2⋅3+3⋅2=191\cdot7+2\cdot3+3\cdot2=19, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?
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