OFFSET
0
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1).
FORMULA
Euler transform of length 12 sequence [ 0, 1, -1, -1, 0, -1, 0, 0, 0, 0, 0, 1].
G.f.: (1 - x^3) / (1 - x^2 + x^4).
a(n) = -(-1)^n * a(n+3) = -a(n+6) = a(-1-n) = a(n+2) - a(n+4) for all n in Z.
EXAMPLE
G.f. = 1 + x^2 - x^3 - x^5 - x^6 - x^8 + x^9 + x^11 + x^12 + x^14 - x^15 + ...
MATHEMATICA
a[ n_] := KroneckerSymbol[ -6, 2 n + 7];
LinearRecurrence[{0, 1, 0, -1}, {1, 0, 1, -1}, 50] (* G. C. Greubel, Jan 15 2018 *)
CoefficientList[Series[(1-x^3)/(1-x^2+x^4), {x, 0, 100}], x] (* Harvey P. Dale, Jun 30 2021 *)
PROG
(PARI) {a(n) = kronecker( -6, 2*n + 7)};
(PARI) {a(n) = (-1)^(n\6 + n) * [1, 0, 1][n%3 + 1]};
(PARI) {a(n) = if( n<0, n=-1-n); polcoeff( (1 - x^3) / (1 - x^2 + x^4) + x * O(x^n), n)};
(Magma) I:=[1, 0, 1, -1]; [n le 4 select I[n] else Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Michael Somos, Jul 18 2015
STATUS
approved