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Sum of two consecutive perfect squares

Source: Romanian IMO Team Selection Test TST 1999, problem 3

September 24, 2005
quadraticsmodular arithmeticinductionfloor functionbinomial coefficientsalgebraspecial factorizations

Problem Statement

Prove that for any positive integer nn, the number Sn=(2n+10)⋅22n+(2n+12)⋅22n−2⋅3+⋯+(2n+12n)⋅3n S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n is the sum of two consecutive perfect squares. Dorin Andrica
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