MathDB

2014-2015 Spring OMO #26

Source:

April 14, 2015

Online Math Open

Problem Statement

Consider a sequence

T_0, T_1, \dots

of polynomials defined recursively by

T_0(x) = 2

,

T_1(x)=x

, and

T_{n+2}(x) = xT_{n+1}(x) - T_n(x)

for each nonnegative integer

n

. Let

L_n

be the sequence of Lucas Numbers, defined by

L_0 = 2

,

L_1 = 1

, and

L_{n+2} = L_n+L_{n+1}

for every nonnegative integer

n

.
Find the remainder when

T_0\left( L_0 \right) + T_1 \left( L_2 \right) + T_2 \left( L_4 \right) + \dots + T_{359} \left( L_{718} \right)

is divided by

359

.
Proposed by Yang Liu

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