cp's OEIS Frontend

A204291 G.f.: Sum_{n>=1} moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where Lucas(n) = A000204(n).

%I A204291 #10 Apr 09 2025 23:39:48
%S A204291 1,0,1,0,4,-3,12,0,17,-10,88,-54,232,-28,184,0,1596,-969,4180,-1230,
%T A204291 4632,-198,28656,-17388,60020,-520,98209,-23604,514228,-461932,
%U A204291 1346268,0,1722688,-3570,6672168,-5598882,24157816,-9348,31351552,-18606210,165580140
%N A204291 G.f.: Sum_{n>=1} moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where Lucas(n) = A000204(n).
%C A204291 Compare g.f. to the identity: x = Sum_{n>=1} moebius(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%H A204291 Paul D. Hanna, <a href="/A204291/b204291.txt">Table of n, a(n) for n = 1..1024</a>
%F A204291 a(k) = 0 iff k = 2^n for n>=1.
%F A204291 a(n) = Fibonacci(n) * Sum_{d|n} mu(d)/Fibonacci(d). - _Ridouane Oudra_, Apr 09 2025
%e A204291 G.f.: A(x) = x + x^3 + 4*x^5 - 3*x^6 + 12*x^7 + 17*x^9 - 10*x^10 + 88*x^11 +...
%e A204291 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^5/(1-11*x^5-x^10) + x^6/(1-18*x^6+x^12) +...+ moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
%p A204291 with(numtheory): seq(add(mobius(d)*combinat[fibonacci](n)/combinat[fibonacci](d), d in divisors(n)), n=1..60); # _Ridouane Oudra_, Apr 09 2025
%o A204291 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A204291 {a(n)=polcoeff(sum(m=1,n,moebius(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%Y A204291 Cf. A203847, A000204 (Lucas), A000045, A008683.
%K A204291 sign
%O A204291 1,5
%A A204291 _Paul D. Hanna_, Jan 14 2012