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binomial sum with mod 7

Source: VJIMC 1996 2.2

October 19, 2021
SummationBinomialbinomial coefficientsnumber theoryalgebra

Problem Statement

Let {xn}n=0∞\{x_n\}^\infty_{n=0} be the sequence such that x0=2x_0=2, x1=1x_1=1 and xn+2x_{n+2} is the remainder of the number xn+1+xnx_{n+1}+x_n divided by 77. Prove that xnx_n is the remainder of the number 4n∑k=0⌊n2⌋2(n2k)5k4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k
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