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