OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, -1).
FORMULA
Euler transform of length 8 sequence [1, -1, 1, -1, 0, -1, 0, 1].
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e otherwise and a(0) = 1.
a(n) = -a(-n) for all n in Z unless n = 0. a(n+4) = -a(n) unless n = 0 or n = -4. a(2*n) = 0 unless n = 0.
a(n) = A188510(n) unless n = 0.
a(n+1) - a(n) = (-1)^n if n>0.
G.f.: (1 + x) * (1 + x^3) / (1 + x^4) = 1 + (x + x^3) / (1 + x^4).
G.f.: (1 - x^2) * (1 - x^4) * (1 - x^6) / ((1 - x) * (1 - x^3) * (1 - x^8)).
G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 - x / (1 + 2*x / (1 - 2*x / (1 - x / (2 + x)))))))).
EXAMPLE
G.f. = 1 + x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 + ...
MATHEMATICA
a[ n_] := If[ EvenQ[ n], Boole[n == 0], (-1)^Quotient[ n, 4]];
a[ n_] := If[ n == 0, 1, Sign[ n] SeriesCoefficient[ (1 + x) * (1 + x^3) / (1 + x^4), {x, 0, Abs @ n}]];
CoefficientList[Series[(1+x)*(1+x^3)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
PROG
(PARI) {a(n) = if( n%2 == 0, n==0, (-1)^(n\4))};
(PARI) {a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x) * (1 + x^3) / (1 + x^4), + x* O(x^abs(n)), abs(n)))};
(PARI) x='x+O('x^60); Vec((1+x)*(1+x^3)/(1+x^4)) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+x^3)/(1+x^4))); // G. C. Greubel, Aug 02 2018
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Michael Somos, Apr 17 2015
STATUS
approved