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),", "))
A209450
a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.
Original entry on oeis.org
1, -3, 6, -15, 36, 0, 210, -1014, 1224, -2955, 0, 0, 41580, -200766, 484692, 0, 1412496, 0, 8232630, -39750654, 0, -231683790, 0, 0, 1630019160, -3935214363, 19000895772, -22936110135, 110745336312, 0, 0, -1558305137094, 1881040698144, 0, 0, 0, 63900011068740
Offset: 0
G.f.: A(x) = 1 - 3*x + 6*x^2 - 15*x^3 + 36*x^4 + 210*x^6 - 1014*x^7 +...
where A(x) = 1 - 1*3*x + 2*3*x^2 - 5*3*x^3 + 12*3*x^4 + 70*3*x^6 - 169*6*x^7 + 408*3*x^8 +...+ Pell(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+2*x-x^2) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 5741*x^11/(1+16238*x^11-x^22) + 33461*x^13/(1+94642*x^13-x^26) - 1136689*x^17/(1+3215042*x^17-x^34) +...).
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A132973[n_]:= SeriesCoefficient[EllipticTheta[2, Pi/4, q^(1/2)]^3/EllipticTheta[2, Pi/4, q^(3/2)]/2, {q, 0, n}]; Join[{1}, Table[ Fibonacci[n, 2]*A132973[n],{n,1,50}]] (* G. C. Greubel, Jan 02 2018 *)
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{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 3*sum(m=0,n, Pell(6*m+1)*x^(6*m+1)/(1+A002203(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - Pell(6*m+5)*x^(6*m+5)/(1+A002203(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ),n)}
for(n=0,61,print1(a(n),", "))
A209452
a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.
Original entry on oeis.org
1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480
Offset: 0
G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 - 6*( 1*x/(1+2*x-x^2) - 2*x^2/(1+6*x^2+x^4) + 12*x^4/(1+34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1+1154*x^8+x^16) + ...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
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A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)
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{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 6*sum(m=1,n,Pell(m)*kronecker(m,3)*x^m/(1+A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))
Showing 1-3 of 3 results.
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