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A225525 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..36.

FORMULA

Logarithmic derivative of A225524 and A203850 (up to sign).

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).

EXAMPLE

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) *...

MATHEMATICA

Table[(DivisorSigma[1, 2n]-DivisorSigma[1, n])LucasL[n], {n, 40}] (* Harvey P. Dale, Sep 10 2016 *)

PROG

(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), ", "))

CROSSREFS

Cf. A203850, A225524.

Sequence in context: A297763 A254962 A139323 * A240395 A009331 A154252

Adjacent sequences:  A225522 A225523 A225524 * A225526 A225527 A225528

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 09 2013

STATUS

approved

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Last modified June 30 19:43 EDT 2022. Contains 354945 sequences. (Running on oeis4.)