A131456 Number of q-partial fraction summands of the reciprocal of n-th cyclotomic polynomial.

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 7, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 1, 2, 1, 2, 7

Offset

1,2

Comments

Let Phi(n,q) be the n-th cyclotomic polynomial in q. The q-partial fraction decomposition of 1/Phi(n,q) is a representation of 1/Phi(n,q) as a finite sum of functions v(q)/(1-q^m)^t, such that m<=n and degree(v)<phi(m) (Euler's totient function A000010).

Links

Table of n, a(n) for n=1..105.

Augustine O. Munagi, Computation of q-partial fractions, INTEGERS: Electronic Journal Of Combinatorial Number Theory, 7 (2007), #A25.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Example

(i) a(3)=1 because 1/Phi(3,q)=(1-q)/(1-q^3);
(ii) a(6)=2 because 1/Phi(6,q)=(-1-q)/(1-q^3) + (2+2q)/(1-q^6).

Crossrefs

Cf. A051664 (Number of terms in n-th cyclotomic polynomial).

Sequence in context: A161263 A161288 A185217 * A074944 A245041 A161315

Adjacent sequences: A131453 A131454 A131455 * A131457 A131458 A131459

Keyword

nonn

Author

Augustine O. Munagi, Jul 12 2007

Status

approved