0,3

The sequence comes from a problem suggested by Fred Lunnon on the math-fun mailing list on Sep 24 2008. a(n) is also the number of polynomials of degree at most n-1 with all coefficients equal to 1 or -1 which are not invertible modulo x^n - 1. I have a proof that a(n) = 2 if n is an odd prime. - W. Edwin Clark

Max Alekseyev's proof that a(2*p) = 2*binomial(2*p,p) if p is an odd prime: First notice that A144926(2p) equals the number of such (1,-1)-polynomials f(x) of degree 2p-1 that are divisible by at least one of the cyclotomic polynomials of orders 1, 2, p, or 2p (that are divisors of 2p). These cyclotomic polynomials are: c(1,x) = x - 1, c(2,x) = x + 1, c(p,x) = x^(p-1) + x^(p-2) + ... + x + 1, c(2p,x) = x^(p-1) - x^(p-2) + ... - x + 1.

Our goal is to prove that i) f(x) may be divisible only by (x-1) or (x+1) but not both ii) if c(p,x) divides f(x) then so does either (x-1) or (x+1) iii) if c(2p,x) divides f(x) then so does either (x-1) or (x+1) They all follow from the following Lemma (that in turn directly follows from the definition of f(x)):

Lemma. The values of f(1) and f(-1) are even but different modulo 4. To prove i), it's enough to notice that if f(x) were divisible by (x - 1) and (x + 1), then f(1) = f(-1) = 0, a contradiction to the Lemma. To prove ii), suppose that f(x) = c(p,x) * q(x) for some integer polynomial q. Then f(1) = p * q(1), together with the Lemma and primality of p implying that f(1) = -2p, 0, or 2p.

But it is easy to see that if f(1)=0, then (x-1) divides f(x); while if f(1) = -2p or 2p, then (x+1) divides f(x). iii) is proved similarly to ii). From i), ii), iii), it follows that to compute A144926(2p) it is enough to compute the number of such f(x) that are divisible by (x-1) and the number of such f(x) that are divisible by (x+1) and add up these numbers. But it is easy to see that each of them equal binomial(2p,p) that completes the proof. [From Vladeta Jovovic (vladeta(AT)eunet.yu), Oct 02 2008]

W. F. Lunnon and Max Alekseyev, Table of n, a(n) for n=1..51

W. F. Lunnon, C program for A144926 and A086328

a(2*n+1) = A086328(2*n+1), n>0. [From Vladeta Jovovic (vladeta(AT)eunet.yu), Sep 29 2008]

If n is an odd prime the only two such matrices are the matrix J with all entries 1 and the matrix -J with all entries -1.

a := proc(n) local T, b, U, M; if isprime(n) and n <> 2 then return 2 end if; T := combinat:-cartprod([seq({-1, 1}, j = 1 .. n)]); b[n] := 0; while not T[finished] do U := T[nextvalue](); M := Matrix(n, shape = Circulant[U]); if LinearAlgebra:-Determinant(M) = 0 then b[n] := b[n]+1 end if end do; return b[n] end proc

Sequence in context: A083489 A065464 A040015 this_sequence A153016 A088610 A026185

Adjacent sequences: A144923 A144924 A144925 this_sequence A144927 A144928 A144929

nonn

W. Edwin Clark (eclark(AT)math.usf.edu), Sep 25 2008; corrected Sep 25 2008

a(22)-a(39) from W. F. Lunnon, Oct 02 2008, Oct 04 2008