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%I #22 Jul 18 2018 06:43:28
%S 1,1,4,3,10,16,26,21,102,110,178,288,466,754,2440,987,3194,7752,8362,
%T 13530,43784,35422,57314,92736,225075,242786,785672,635622,1028458,
%U 3328160,2692538,2178309,14098312,11405774,36909860,44791056,48315634,78176338,252983944
%N a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.
%C Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).
%H G. C. Greubel, <a href="/A205965/b205965.txt">Table of n, a(n) for n = 1..2500</a>
%F G.f.: Sum_{n>=1} Fibonacci(2*n-1)*x^(2*n-1)/(1 - Lucas(2*n-1)*x^(2*n-1)-x^(4*n-2)).
%e G.f.: A(x) = x + x^2 + 4*x^3 + 3*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 21*x^8 +...
%e where A(x) = 1*1*x + 1*1*x^2 + 2*2*x^3 + 3*1*x^4 + 5*2*x^5 + 8*2*x^6 + 13*2*x^7 + 21*1*x^8 +...+ Fibonacci(n)*A001227(n)*x^n +...
%e The g.f. is also given by the identity:
%e A(x) = 1*x/(1-x-x^2) + 2*x^3/(1-4*x^3-x^6) + 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) + 34*x^9/(1-76*x^9-x^18) + 89*x^11/(1-199*x^11-x^22) +...
%e which involves odd-indexed Fibonacci and Lucas numbers.
%t A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* _G. C. Greubel_, Jul 17 2018 *)
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=polcoeff(sum(m=1,n,fibonacci(2*m-1)*x^(2*m-1)/(1-Lucas(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
%o for(n=1,40,print1(a(n),", "))
%o (PARI) a(n) = fibonacci(n)*sumdiv(n, d, d%2); \\ _Michel Marcus_, Jul 18 2018
%Y Cf. A001227, A205964, A205966, A203847, A000204 (Lucas).
%Y Cf. A209445 (Pell variant).
%K nonn
%O 1,3
%A _Paul D. Hanna_, Feb 03 2012
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