A140666 A triangle of coefficients of the difference of prime cyclotomic doubled polynomials: p(x,n)=(c(x,Prime[n])-c(x,2*Prime[n]))/x.

1, -1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2

Offset

1,3

Comments

Row sums are: {0, 2, 4, 6, 10, 12, 16, 18, 22, 28, ...}

The factor x is used instead of 2x to get an integer n=1 term.

p(x,n)/2 are related to the double product:two primes n,m such that

Cyclotomic[Prime[n], x]* Cyclotomic[2*Prime[n]=(Cyclotomic[Prime[m], x] - Cyclotomic[2*Prime[m], x])/(2*x)

Links

Table of n, a(n) for n=1..111.

Formula

c(x,n)=CyclotomicPolynomial; c(x,Prime[n])=(x^Prime[n]-1)/(x-1); p(x,n)=(c(x,Prime[n])-c(x,2*Prime[n]))/x; a(n,m)=Coefficients(p(x,n))

Example

{1, -1},
{2},
{2, 0, 2},
{2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},
{2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2}

Mathematica

Clear[p, x, n] p[x_, n_] = (Cyclotomic[Prime[n], x] - Cyclotomic[2*Prime[n], x])/x; Table[ExpandAll[p[x, n]], {n, 1, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[a]

Crossrefs

Sequence in context: A215879 A114700 A353768 * A350628 A202145 A130772

Adjacent sequences: A140663 A140664 A140665 * A140667 A140668 A140669

Keyword

tabf,uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Jul 11 2008

Status

approved