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A254011 Expansion of (1 - x^18) / ((1 - x^5) * (1 - x^6) * (1 - x^9)) in powers of x. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A002635 A275806 A228369

Adjacent sequences:  A254008 A254009 A254010 * A254012 A254013 A254014

KEYWORD

nonn

AUTHOR

Michael Somos, Jan 22 2015

STATUS

approved

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Last modified January 23 03:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)