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A205967
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a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.
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6
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1, 12, 36, 24, 252, 360, 288, 1248, 3780, 408, 11880, 12816, 12096, 39144, 108576, 43920, 367164, 344952, 93024, 1003440, 3409560, 1050816, 7651152, 8253216, 8346240, 27909300, 61182072, 2357016, 213568992, 185122440, 179720640, 516967296, 1646801604, 507539232
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OFFSET
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0,2
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COMMENTS
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Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).
Here Chi(n,3) = principal Dirichlet character of n modulo 3.
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LINKS
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FORMULA
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G.f.: 1 + 12*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
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EXAMPLE
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G.f.: A(x) = 1 + 12*x + 36*x^2 + 24*x^3 + 252*x^4 + 360*x^5 + 288*x^6 +...
where A(x) = 1 + 1*12*x + 1*36*x^2 + 2*12*x^3 + 3*84*x^4 + 5*72*x^5 + 8*36*x^6 +...+ Fibonacci(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 13*7*x^7/(1-29*x^7-x^14) + 21*8*x^8/(1-47*x^8-x^16) +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].
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MATHEMATICA
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terms = 34; s = 1 + 12*Sum[Fibonacci[n]*KroneckerSymbol[n, 3]^2*n*(x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n))), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
b[n_] := If[n < 1, Boole[n == 0], 12 Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Table[If[n == 0, 1, b[n]*Fibonacci[n]], {n, 0, 50}] (* G. C. Greubel, Jul 17 2018 *)
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PROG
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(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 12*sum(m=1, n, fibonacci(m)*kronecker(m, 3)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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