WOLFRAM|DEMONSTRATIONS PROJECT

Factoring the Even Trigonometric Polynomials of A269254

​
show
x
k
y
k
factorization over 
i
wavefunction decomposition
V
1
|
′
M
|
V
u
: constraint check
M
u
-1
M
u
V
1
check
all k×K elements
reduce modulo period 2k+1
permute columns modulo p
prime p
2
3
5
7
11
13
17
19
23
29
index k
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1 +

2
+

4
+

6
=
(1 +

1
+

2
+

3
) × (-1 +

1
-

2
+

3
)
From reference [2] comes a question regarding the linear recurrence of A269254 [1],
s
k
=n
s
k-1
-
s
k-2
with
s
0
=1,
s
1
=n+1
; let
a(n)
be the smallest index
k
such that
s(k)
is prime, or
-1
if no such
k
exists. For what
n
does the sequence visit
-1
?
Andrew Hone noticed that
a(n)=-1
seems only to occur when
n=

p
(j)
for prime
p
,
j>2
and

p
(j)
is a trigonometric polynomial of order
p
[3] (see the details for more definitions). In these cases, we have proven that the solution
z
k
of the recurrence
z
k
=

p
(j)
z
k-1
-
z
k-2
with
z
0
=1,
z
1
=1+

p
(j)
can be written as a simple summation,
z
k
=1+
k
∑
i=1

pi
(j)
. This Demonstration gives an algorithm that uses linear algebra to calculate the polynomial factorization of
s
k
=
z
k
in all cases
n=

p
(j)
for prime
p
and
j>2
. This Demonstration helps to extend earlier empirical observations and proofs for cases
p=2,3
[4, 5, 6].