A203847
a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.
Original entry on oeis.org
1, 2, 4, 9, 10, 32, 26, 84, 102, 220, 178, 864, 466, 1508, 2440, 4935, 3194, 15504, 8362, 40590, 43784, 70844, 57314, 370944, 225075, 485572, 785672, 1906866, 1028458, 6656320, 2692538, 13069854, 14098312, 22811548, 36909860, 134373168, 48315634, 156352676, 252983944
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...
where A(x) = x/(1-x-x^2) + x^2/(1-3*x^2+x^4) + 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) + 5*x^5/(1-11*x^5-x^10) + 8*x^6/(1-18*x^6+x^12) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
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Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)
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{a(n)=sigma(n,0)*fibonacci(n)}
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
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a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018
A205971
a(n) = Fibonacci(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.
Original entry on oeis.org
1, 4, 4, 8, 60, 120, 32, 416, 1092, 136, 1320, 4272, 2880, 13048, 12064, 14640, 114492, 114984, 10336, 334480, 811800, 350272, 850128, 2751072, 2411136, 9303100, 6798008, 785672, 50849760, 61707480, 19968960, 172322432, 531507396, 169179744, 410607864
Offset: 0
G.f.: A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 60*x^4 + 120*x^5 + 32*x^6 + ...
where A(x) = 1 + 1*4*x + 1*4*x^2 + 2*4*x^3 + 3*20*x^4 + 5*24*x^5 + 8*4*x^6 + ... + Fibonacci(n)*A034896(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 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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A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n]*A034896[n], {n, 1, 50}]] (* G. C. Greubel, Dec 24 2017 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 4*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=0,61,print1(a(n),", "))
A205884
a(n) = Pell(n)*A109064(n) for n >= 1 with a(0)=1.
Original entry on oeis.org
1, -5, 10, 50, -180, -145, -700, 5070, 10200, -34475, 11890, -344460, 415800, 2007660, -2423460, 1950250, -25895760, 90935120, 96047350, -662510900, -239916420, -2316837900, 5593341480, 24756454910, -27166986000, -6558690605, -190008957720, 764537004500
Offset: 0
G.f.: A(x) = 1 - 5*x + 10*x^2 + 50*x^3 - 180*x^4 - 145*x^5 - 700*x^6 + ...
where A(x) = 1 - 1*5*x + 2*5*x^2 + 5*10*x^3 - 12*15*x^4 - 29*5*x^5 - 70*10*x^6 + 169*30*x^7 + 408*25*x^8 + ... + Pell(n)*A109064(n)*x^n + ...
The g.f. is illustrated by:
A(x) = 1 - 5*(+1)*1*1*x/(1-2*x-x^2) - 5*(-1)*2*2*x^2/(1-6*x^2+x^4) - 5*(-1)*3*5*x^3/(1-14*x^3-x^6) - 5*(+1)*4*12*x^4/(1-34*x^4+x^8) - 5*(0)*5*29*x^5/(1-82*x^5-x^10) - 5*(+1)*6*70*x^6/(1-198*x^6+x^12) + ...
The values of the Legendre symbol L(n,5) repeat: [1,-1,-1,1,0, ...].
The companion Pell numbers (A002203) begin: [2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,...].
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pell[n_] := ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2]) // Simplify; (* b = A109064 *); b[0] = 1; b[n_] := b[n] = Sum[DivisorSum[j, #*If[Divisible[#, 5], -4, -5] &]*b[n - j], {j, 1, n}]/n; a[0] = 1; a[n_] := pell[n]*b[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2017 *)
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{A109064(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^5/eta(x^5+A), n))}
{a(n)=if(n==0,1,Pell(n)*A109064(n))}
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{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=polcoeff(1-5*sum(m=1, n, kronecker(m, 5)*m*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
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