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A203802 a(n) = Fibonacci(n) * Sum_{d|n} -(-1)^(n/d) / Fibonacci(d).

%I A203802 #11 Aug 20 2019 05:03:13
%S A203802 1,0,3,-5,6,-3,14,-48,52,-10,90,-329,234,-28,1038,-2349,1598,-1044,
%T A203802 4182,-12750,17262,-198,28658,-135285,90031,-520,300405,-554974,
%U A203802 514230,-464658,1346270,-5188656,5326470,-3570,11782764,-34556612,24157818,-9348,95140422,-256249218,165580142
%N A203802 a(n) = Fibonacci(n) * Sum_{d|n} -(-1)^(n/d) / Fibonacci(d).
%H A203802 Paul D. Hanna, <a href="/A203802/b203802.txt">Table of n, a(n) for n = 1..1000</a>
%F A203802 G.f.: Sum_{n>=1} x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).
%e A203802 G.f.: A(x) = x + 3*x^3 - 5*x^4 + 6*x^5 - 3*x^6 + 14*x^7 - 48*x^8 + 52*x^9 +...
%e A203802 where A(x) = x/(1+x-x^2) + x^2/(1+3*x^2+x^4) + x^3/(1+4*x^3-x^6) + x^4/(1+7*x^4+x^8) + x^5/(1+11*x^5-x^10) + x^6/(1+18*x^6+x^12) +...+ x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
%e A203802 Illustration of terms.
%e A203802 a(1) = 1; a(2) = 1*(-1/1 + 1/1) = 0; a(3) = 2*(1/1 + 1/2) = 3;
%e A203802 a(4) = 3*(-1/1 - 1/1 + 1/3) = -5; a(5) = 5*(1/1 + 1/5) = 6;
%e A203802 a(6) = 8*(-1/1 + 1/1 - 1/2 + 1/8) = -3; a(7) = 13*(1/1 + 1/13) = 14;
%e A203802 a(8) = 21*(-1/1 - 1/1 - 1/3 + 1/21) = -48; ...
%o A203802 (PARI) {a(n)=fibonacci(n) * sumdiv(n, d, -(-1)^(n/d) / fibonacci(d))}
%o A203802 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203802 {a(n)=polcoeff(sum(m=1, n, x^m/(1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}
%Y A203802 Cf. A111075, A000045, A000204 (Lucas).
%K A203802 sign
%O A203802 1,3
%A A203802 _Paul D. Hanna_, Jan 11 2012