OFFSET
0,2
COMMENTS
Also continued fraction expansion of sqrt(2717)/38. - Bruno Berselli, Mar 07 2011
LINKS
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).
FORMULA
Euler transform of length 8 sequence [ 2, -2, 2, 0, 0, -2, 0, 1].
Moebius transform is length 8 sequence [ 2, -1, 0, 3, 0, 0, 0, -2].
a(n) = 2 * b(n) where b() is multiplicative with b(2) = 1/2, b(4) = 2, b(2^e) = 1 if e>2, b(p^e) = 1 if p>2.
G.f.: (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) = (1-x+x^2)^2*(1+x)^3 / ((1-x) *(1+x^2) *(1+x^4)). a(-n) = a(n). a(2*n + 1) = 2, a(4*n + 2) = 1, a(8*n + 4) = 4, a(8*n) = 2 except a(0) = 1.
EXAMPLE
1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + ...
MATHEMATICA
PadRight[{1}, 108, {2, 2, 1, 2, 4, 2, 1, 2}] (* Harvey P. Dale, Mar 22 2012 *)
PROG
(PARI) {a(n) = - (n==0) + [ 2, 2, 1, 2, 4, 2, 1, 2] [n%8 + 1]}
(PARI) {a(n) = polcoeff( (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) + x * O(x^abs(n)), abs(n))}
(Magma) [1] cat &cat[ [2, 1, 2, 4, 2, 1, 2, 2]: n in [1..13]]; // Bruno Berselli, Mar 07 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Feb 14 2011
STATUS
approved