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
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 22 2015
STATUS
approved