cp's OEIS Frontend

A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).

%I A203853 #19 Dec 26 2017 03:15:28
%S A203853 1,4,5,10,24,50,120,270,640,1500,3600,8610,20880,50700,124024,304290,
%T A203853 750120,1854400,4600200,11440548,28527320,71289000,178526880,
%U A203853 447910470,1125750120,2833885800,7144449920,18036373140,45591631800,115381697740,292329067800,741410800830
%N A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).
%C A203853 Apparently the same as A032170, if n > 2. - _R. J. Mathar_, Jan 11 2012
%H A203853 Paul D. Hanna, <a href="/A203853/b203853.txt">Table of n, a(n) for n = 1..640</a>
%F A203853 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^3 * x^n/n), which is the g.f. of A203803.
%F A203853 a(n) ~ phi^(2*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Sep 02 2017
%e A203853 G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^4 * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^10 * (1-11*x^5-x^10)^24 * (1-18*x^6+x^12)^50 * (1-29*x^7-x^14)^120 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
%e A203853 where F(x) = exp( Sum_{n>=1} Lucas(n)^3 * x^n/n ) = g.f. of A203803:
%e A203853 F(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 + ...
%e A203853 where
%e A203853 log(F(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 + ... + Lucas(n)^3*x^n/n + ...
%t A203853 a[n_]:= 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^2 &]; Array[a, 30] (* _G. C. Greubel_, Dec 25 2017 *)
%o A203853 (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^2)/n)}
%o A203853 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203853 {a(n)=local(F=exp(sum(m=1, n, Lucas(m)^3*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}
%Y A203853 Cf. A001254, A203803, A006206, A203854, A203855, A203856, A203857, A203858, A203859, A203800.
%K A203853 nonn
%O A203853 1,2
%A A203853 _Paul D. Hanna_, Jan 07 2012