A254011 Expansion of (1 - x^18) / ((1 - x^5) * (1 - x^6) * (1 - x^9)) in powers of x.

1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 3, 4, 5, 4, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 4, 5, 6, 5, 5, 5, 5, 6, 6, 5, 5, 6, 6, 6

Offset

0,16

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Tom Fisher, Invariant theory for the elliptic normal quintic, I. Twists of X(5), page 10.

Tom Fisher, Invariant theory for the elliptic normal quintic, I. Twists of X(5), arXiv:1110.3520 [math.NT], 2011, page 10.

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).

Formula

Euler transform of length 18 sequence [ 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1].

G.f.: (1 - x^3 + x^6) / (1 - x^3 - x^5 + x^8) = (1 - x^3 + x^6) / ( (1 - x)^2 * (1 + x + x^2) * (1 + x + x^2 + x^3 + x^4)).

a(n) = -a(-2-n), a(n+15) = 1 + a(n), for all n in Z.

0 = a(n) - a(n+3) - a(n+5) + a(n+8) for all n in Z.

a(5*n) = A008620(n). a(5*n + 1) = a(5*n + 4) = A008620(n-1). a(5*n + 2) = A008620(n-2). a(5*n + 3) = A008620(n-3).

0 = -1 + a(n)*(+a(n) - a(n+1) - 2*a(n+3) + a(n+4)) +a(n+1)*(+a(n+1) + a(n+3) - 2*a(n+4)) +a(n+3)*(+a(n+3) - a(n+4)) +a(n+4)*(+a(n+4)) for all n in Z.

Example

G.f. = 1 + x^5 + x^6 + x^9 + x^10 + x^11 + x^12 + x^14 + 2*x^15 + x^16 + ...
G.f. = q + q^11 + q^13 + q^19 + q^21 + q^23 + q^25 + q^29 + 2*q^31 + q^33 + ...

Mathematica

CoefficientList[Series[(1-x^3+x^6)/(1-x^3-x^5+x^8), {x, 0, 60}], x] (* G. C. Greubel, Aug 04 2018 *)
LinearRecurrence[{0, 0, 1, 0, 1, 0, 0, -1}, {1, 0, 0, 0, 0, 1, 1, 0}, 90] (* Harvey P. Dale, Apr 30 2019 *)

Prog

(PARI) {a(n) = my(m=n%15); (n+6) \ 15 + (m==0) + (m==5) + (m==6) - (m==13)};
(PARI) {a(n) = n++; sign(n) * polcoeff( x * (1 - x^3 + x^6) / (1 - x^3 - x^5 + x^8) + x * O(x^abs(n)), abs(n))};
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^3+x^6)/(1-x^3-x^5+x^8))); // G. C. Greubel, Aug 04 2018

Crossrefs

Cf. A008620.

Sequence in context: A001876 A033182 A053797 * A361919 A002635 A275806

Adjacent sequences: A254008 A254009 A254010 * A254012 A254013 A254014

Keyword

nonn

Author

Michael Somos, Jan 22 2015

Status

approved