|
| |
|
|
A163806
|
|
Expansion of (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^6)) in powers of x.
|
|
3
|
|
|
|
1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
Euler transform of length 6 sequence [1, -1, -1, 0, 0, 1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (2*v - 1) - (3*v - 2) * (2*u - 1).
a(n) is multiplicative with a(3^e) = 0^e, a(2^e) = -(-1)^e if e>0, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
a(3*n) = 0 unless n=0, a(6*n + 4) = a(6*n + 5) = -1, a(6*n + 1) = a(6*n + 2) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.
G.f.: (1 + x^2) / (1 - x + x^2). - Corrected by Bruno Berselli, Apr 06 2011
G.f.: A(x) = 1 / (1 - x / (1 + x^2)) = 1 + x / (1 - x / (1 + x / (1 - x))). - Michael Somos, Jan 03 2013
|
|
|
EXAMPLE
|
G.f. = 1 + x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 + x^14 + ...
|
|
|
MATHEMATICA
|
Join[{1}, LinearRecurrence[{1, -1}, {1, 1}, 104]] (* Ray Chandler, Sep 15 2015 *)
|
|
|
PROG
|
(PARI) {a(n) = (n==0) + [0, 1, 1, 0, -1, -1][n%6 + 1]};
(PARI) {a(n) = (n==0) - (-1)^n * kronecker(-3, n)};
(Magma) I:=[1, 1]; [1] cat [n le 2 select I[n] else Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign,easy,mult
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
