# Factoring the Even Trigonometric Polynomials of A269254

Factoring the Even Trigonometric Polynomials of A269254

From reference [2] comes a question regarding the linear recurrence of A269254 [1], =n- with =1,=n+1; let be the smallest index such that is prime, or if no such exists. For what does the sequence visit ?

s

k

s

k-1

s

k-2

s

0

s

1

a(n)

k

s(k)

-1

k

n

-1

Andrew Hone noticed that seems only to occur when for prime , and (j) is a trigonometric polynomial of order [3] (see the details for more definitions). In these cases, we have proven that the solution of the recurrence =(j)- with =1,=1+(j) can be written as a simple summation, =1+(j). This Demonstration gives an algorithm that uses linear algebra to calculate the polynomial factorization of = in all cases for prime and . This Demonstration helps to extend earlier empirical observations and proofs for cases [4, 5, 6].

a(n)=-1

n=(j)

p

p

j>2

p

p

z

k

z

k

p

z

k-1

z

k-2

z

0

z

1

p

z

k

k

∑

i=1

pi

s

k

z

k

n=(j)

p

p

j>2

p=2,3