MathDB

Problem 2

Part of 1996 VJIMC

Problems(2)

binomial sum with recurrence relation

Source: VJIMC 1996 1.2

10/12/2021
Let {an}n=0∞\{a_n\}^\infty_{n=0} be the sequence of integers such that a0=1a_0=1, a1=1a_1=1, an+2=2an+1−2ana_{n+2}=2a_{n+1}-2a_n. Decide whether an=∑k=0⌊n2⌋(n2k)(−1)k.a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.
Sequencesrecurrence relationSummationBinomialalgebra
binomial sum with mod 7

Source: VJIMC 1996 2.2

10/19/2021
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
SummationBinomialbinomial coefficientsnumber theoryalgebra