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A225525
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a(n) = (sigma(2*n) - sigma(n))*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.
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2
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2, 12, 32, 56, 132, 288, 464, 752, 1976, 2952, 4776, 10304, 14588, 26976, 65472, 70624, 128556, 300456, 373960, 726096, 1566464, 1900944, 3075792, 6635648, 10401182, 15200808, 35136320, 45481408, 68991060, 178607808, 192662336, 311734208, 756594816, 918147096, 1980790944, 3472069328
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OFFSET
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1,1
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COMMENTS
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Compare l.g.f. to log(theta_4(x)) = Sum_{n>=1} (sigma(2*n)-sigma(n))*x^n/n, where Jacobi theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2).
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LINKS
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FORMULA
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L.g.f.: Sum_{n>=1} log( (1 + Lucas(n)*x^n + (-x^2)^n) / (1 - Lucas(n)*x^n + (-x^2)^n) ) = Sum_{n>=1} a(n)*x^n/n.
a(n) == 0 (mod 2); a(n) == 2 (mod 4) iff n is congruent to 1 or 5 mod 6 (A007310).
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EXAMPLE
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L.g.f.: L(x) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...
where
exp(-L(x)) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...+ A203850(n)*x^n +...
Also,
exp(L(x)) = 1 + 2*x + 8*x^2 + 24*x^3 + 66*x^4 + 184*x^5 + 488*x^6 + 1248*x^7 +...+ A225524(n)*x^n +...
Exponentiation yields the product:
exp(L(x)) = (1+x-x^2)/(1-x-x^2) * (1+3*x^2+x^4)/(1-3*x^2+x^4) * (1+4*x^3-x^6)/(1-4*x^3-x^6) * (1+7*x^4+x^8)/(1-7*x^4+x^8) * (1+11*x^5-x^10)/(1-11*x^5-x^10) *...* (1 + Lucas(n)*x^n + (-x^2)^n)/(1 - Lucas(n)*x^n + (-x^2)^n) *...
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MATHEMATICA
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Table[(DivisorSigma[1, 2n]-DivisorSigma[1, n])LucasL[n], {n, 40}] (* Harvey P. Dale, Sep 10 2016 *)
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PROG
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(PARI) {a(n)=(sigma(2*n) - sigma(n))*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1, 40, print1(a(n), ", "))
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(-log(prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n)))), n)}
for(n=1, 40, 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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