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A205507
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a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.
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5
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1, 4, 4, 0, 12, 40, 0, 0, 84, 136, 440, 0, 0, 1864, 0, 0, 3948, 12776, 10336, 0, 54120, 0, 0, 0, 0, 900300, 971144, 0, 0, 4113832, 0, 0, 8713236, 0, 45623096, 0, 59721408, 193262536, 0, 0, 818673240, 1324641128, 0, 0, 0, 9079225360, 0, 0, 0, 31114968196
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OFFSET
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0,2
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COMMENTS
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Compare to the g.f. of A004018 given by the Lambert series identity:
1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.
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LINKS
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FORMULA
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G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Fibonacci(2*n+1)*x^(2*n+1) / (1 - Lucas(2*n+1)*x^(2*n+1) - x^(4*n+2)), where Lucas(n) = A000204(n).
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 4*x^2 + 12*x^4 + 40*x^5 + 84*x^8 + 136*x^9 + 440*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-x-x^2) - 4*2*x^3/(1-4*x^3-x^6) + 4*5*x^5/(1-11*x^5-x^10) - 4*13*x^7/(1-29*x^7-x^14) + 4*34*x^9/(1-76*x^9-x^18) - 4*89*x^11/(1-199*x^11-x^22) + 4*233*x^13/(1-521*x^13-x^26) - 4*610*x^15/(1-1364*x^15-x^30) +...
which involves odd-indexed Fibonacci and Lucas numbers.
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MATHEMATICA
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Join[{1}, Table[Fibonacci[n]*SquaresR[2, n], {n, 1, 50}]] (* G. C. Greubel, Mar 05 2017 *)
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PROG
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(PARI) {A004018(n)=polcoeff((1+2*sum(k=1, sqrtint(n+1), x^(k^2), x*O(x^n)))^2, n)}
{a(n)=if(n==0, 1, fibonacci(n)*A004018(n))}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff((1+4*sum(m=0, n+1, (-1)^m*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)}
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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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