%I #18 Sep 08 2022 08:46:11
%S 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,
%T 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,
%U 4,5,5,5,5,4,5,6,5,5,5,5,6,6,5,5,6,6,6
%N Expansion of (1 - x^18) / ((1 - x^5) * (1 - x^6) * (1 - x^9)) in powers of x.
%H G. C. Greubel, <a href="/A254011/b254011.txt">Table of n, a(n) for n = 0..2500</a>
%H Tom Fisher, <a href="http://www.dpmms.cam.ac.uk/~taf1000/papers/invenqI.html">Invariant theory for the elliptic normal quintic, I. Twists of X(5)</a>, page 10.
%H Tom Fisher, <a href="https://arxiv.org/abs/1110.3520">Invariant theory for the elliptic normal quintic, I. Twists of X(5)</a>, arXiv:1110.3520 [math.NT], 2011, page 10.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,1,0,0,-1).
%F Euler transform of length 18 sequence [ 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1].
%F 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)).
%F a(n) = -a(-2-n), a(n+15) = 1 + a(n), for all n in Z.
%F 0 = a(n) - a(n+3) - a(n+5) + a(n+8) for all n in Z.
%F 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).
%F 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.
%e G.f. = 1 + x^5 + x^6 + x^9 + x^10 + x^11 + x^12 + x^14 + 2*x^15 + x^16 + ...
%e G.f. = q + q^11 + q^13 + q^19 + q^21 + q^23 + q^25 + q^29 + 2*q^31 + q^33 + ...
%t CoefficientList[Series[(1-x^3+x^6)/(1-x^3-x^5+x^8), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 04 2018 *)
%t LinearRecurrence[{0,0,1,0,1,0,0,-1},{1,0,0,0,0,1,1,0},90] (* _Harvey P. Dale_, Apr 30 2019 *)
%o (PARI) {a(n) = my(m=n%15); (n+6) \ 15 + (m==0) + (m==5) + (m==6) - (m==13)};
%o (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))};
%o (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
%Y Cf. A008620.
%K nonn
%O 0,16
%A _Michael Somos_, Jan 22 2015
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