A000204 -id:A000204 - cp's OEIS frontend

A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

1, 1, 365, 1730, 97390, 948562, 26292937, 370813165, 7716851405, 127699557640, 2397734250216, 42004273130216, 763345960355450, 13608990417046650, 245008471017094450, 4389301146029065420, 78826300825689660420, 1413927351334191841100, 25376664633745265522450

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

			G.f.: A(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
where
log(A(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..795
Index entries for linear recurrences with constant coefficients, order 64.

Cf. A002571, A203803, A203804, A203805, A203807, A203808, A203809.

Cf. A203856, A203800.

CoefficientList[Series[1/((1 + x)^20*(1 - 3*x + x^2)^15*(1 + 7*x + x^2)^6*(1 - 18*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^6*x^k/k)+x*O(x^n)), n)}

{a(n,m=3)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

G.f.: 1/( (1+x)^20 * (1-3*x+x^2)^15 * (1+7*x+x^2)^6 * (1-18*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203856(n) where A203856(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^5.

A156216 G.f.: A(x) = exp( Sum_{n>=1} A000204(n)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 26, 634, 32928, 5704263, 2470113915, 2978904483553, 9401949327631932, 79268874871208384494, 1762019469678472912173354, 103537245443913551792800303420, 16030602885085486700462431379649694

Offset: 0

Paul D. Hanna, Feb 06 2009

nonn

Compare to g.f. of Fibonacci sequence: exp( Sum_{n>=1} A000204(n)*x^n/n ), where A000204 is the Lucas numbers.

More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture).

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 +...

Cf. A000045, A000204, A156212.

Cf. A067961. [From Paul D. Hanna, Sep 13 2010]

{a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m+1)+fibonacci(m-1))^m*x^m/m)+x*O(x^n)),n)}

a(n) = (1/n)*Sum_{k=1..n} A000204(k)^k*a(n-k) for n>0, with a(0) = 1.

Logarithmic derivative forms A067961. [From Paul D. Hanna, Sep 13 2010]

A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 122, 463, 11985, 85456, 1262166, 12018742, 145326748, 1540766090, 17495016342, 191731126832, 2138972609189, 23652975370501, 262682339212290, 2911255335387883, 32296421465575573, 358120616523262016, 3971885483375619384, 44047530724737577400

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
where
log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (1,121,220,-3460,-5932,52717,52667,-483925,-81600,2532240,-1172640,-6764090,4911050,11191850,-8809960,-13039640,8809960,11191850,-4911050,-6764090,1172640,2532240,81600,-483925,-52667,52717,5932,-3460,-220,121,-1,-1).

Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.

Cf. A203855, A203800.

CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}

{a(n,m=2)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.

A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 1094, 6555, 809765, 10676072, 570282082, 11680775298, 427757608420, 10880625876510, 341910837405634, 9500984180929624, 282684350289144641, 8100555748749977985, 236841648715969283630, 6851665210550903756723, 199305150210062939465293

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where
log(A(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..680
Index entries for linear recurrences with constant coefficients, order 128.

Cf. A002571, A203803, A203804, A203805, A203806, A203808, A203809.

Cf. A203857, A203800.

CoefficientList[Series[1/((1 + x - x^2)^35*(1 - 4*x - x^2)^21*(1 + 11*x - x^2)^7*(1 - 29*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^7*x^k/k)+x*O(x^n)), n)}

{a(n,m=3)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

G.f.: 1/( (1+x-x^2)^35 * (1-4*x-x^2)^21 * (1+11*x-x^2)^7 * (1-29*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203857(n) where A203857(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^6.

A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 3281, 25126, 6845526, 121368902, 12805025677, 373879862237, 24707348223677, 948781359159752, 50702478932197928, 2210812262034197128, 108528095366637700218, 4974402150387759436378, 236926456045384849970778, 11047772769135934828000404

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

			G.f.: A(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(A(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..596
Index entries for linear recurrences with constant coefficients, order 256.

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203809.

Cf. A203858, A203800.

CoefficientList[Series[1/((1 - x)^70*(1 + 3*x + x^2)^56*(1 - 7*x + x^2)^28*(1 + 18*x + x^2)^8*(1 - 47*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^8*x^k/k)+x*O(x^n)), n)}

{a(n,m=4)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

G.f.: 1/( (1-x)^70 * (1+3*x+x^2)^56 * (1-7*x+x^2)^28 * (1+18*x+x^2)^8 * (1-47*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203858(n) where A203858(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^7.

A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 4, 5, 10, 24, 50, 120, 270, 640, 1500, 3600, 8610, 20880, 50700, 124024, 304290, 750120, 1854400, 4600200, 11440548, 28527320, 71289000, 178526880, 447910470, 1125750120, 2833885800, 7144449920, 18036373140, 45591631800, 115381697740, 292329067800, 741410800830

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

Apparently the same as A032170, if n > 2. - R. J. Mathar, Jan 11 2012

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^4 * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^10 * (1-11*x^5-x^10)^24 * (1-18*x^6+x^12)^50 * (1-29*x^7-x^14)^120 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^3 * x^n/n ) = g.f. of A203803:
F(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 + ...
where
log(F(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 + ... + Lucas(n)^3*x^n/n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..640

Cf. A001254, A203803, A006206, A203854, A203855, A203856, A203857, A203858, A203859, A203800.

a[n_]:= 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^2 &]; Array[a, 30] (* G. C. Greubel, Dec 25 2017 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^2)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^3*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^3 * x^n/n), which is the g.f. of A203803.

a(n) ~ phi^(2*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A304333 Number of positive integers k such that n - L(k) is a positive squarefree number, where L(k) denotes the k-th Lucas number A000204(k).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 5, 2, 3, 4, 5, 2, 4, 4, 4, 3, 5, 4, 4, 2, 3, 3, 5, 3, 5, 5, 5, 4, 4, 5, 4, 4, 6, 5, 6, 3, 6, 4, 5, 3, 6, 5, 6, 3, 5, 4, 5, 3, 3, 4, 6, 4, 6, 4, 7, 3, 6, 4, 6, 2, 6, 6, 6, 4, 5, 6, 4, 4, 6, 7, 6, 3, 7, 6, 6, 4, 6, 5, 7, 5, 6, 7, 8

Offset: 1

Zhi-Wei Sun, May 11 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 5*10^9.
See also A304331 for a similar conjecture involving Fibonacci numbers.
For all n, a(n) <= A130241(n). - Antti Karttunen, May 13 2018

			a(2) = 1 with 2 - L(1) = 1 squarefree.
a(3) = 1 with 3 - L(1) = 2 squarefree.
a(67) = 2 with 67 - L(1) = 2*3*11 and 67 - L(7) = 2*19 both squarefree.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000032, A000204, A005117, A102460, A130241, A304034, A304081, A304331.

a := proc(n) local count, lucas, newcas;
count := 0; lucas := 1; newcas := 2;
while lucas < n do
    if numtheory:-issqrfree(n - lucas) then count := count + 1 fi;
    lucas, newcas := lucas + newcas, lucas;
od;
count end:
seq(a(n), n=1..90); # Peter Luschny, May 15 2018

f[n_]:=f[n]=LucasL[n];
tab={};Do[r=0;k=1;Label[bb];If[f[k]>=n,Goto[aa]];If[SquareFreeQ[n-f[k]],r=r+1];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,90}];Print[tab]

A304333(n) = { my(u1=1,u2=3,old_u1,c=0); if(n<=2,n-1,while(u1Antti Karttunen, May 13 2018

A067592 Number of partitions of n into Lucas parts (A000204(k) for k >= 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 6, 7, 8, 10, 13, 15, 17, 21, 25, 28, 32, 39, 44, 49, 57, 66, 73, 82, 94, 105, 116, 130, 147, 162, 178, 199, 221, 241, 265, 295, 322, 350, 385, 423, 458, 498, 545, 592, 639, 693, 755, 814, 876, 949, 1026, 1100, 1183, 1278, 1371, 1467, 1576, 1694, 1809, 1933, 2072, 2215, 2359, 2517, 2691

Offset: 0

Naohiro Nomoto, Jan 31 2002

			a(7) counts these partitions: 7, 43, 4111, 331, 31111, 1111111. - _Clark Kimberling_, Mar 08 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

p[n_] := IntegerPartitions[n, All, LucasL@Range@15]; Table[p[n], {n, 0, 12}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0,80] (* counts partitions, A067592 *)
(* Clark Kimberling, Mar 08 2014 *)
Table[SeriesCoefficient[gf = 1; k = 1; While[LucasL[k] <= n, gf = gf*(1 - x^LucasL[k]); k++]; gf = 1/gf, {x, 0, n}], {n, 0, 100}] (* Vaclav Kotesovec, Mar 26 2014, after Joerg Arndt *)

N=66; q='q+O('q^N);
L(n) = fibonacci(n+2) - fibonacci(n-2);
gf = 1; k=1; while( L(k) <= N, gf*=(1-q^L(k)); k+=1 ); gf = 1/gf;
Vec( gf ) /* Joerg Arndt, Mar 26 2014 */

G.f.: 1/Product_{n>=1} (1 - q^A000204(n)). - Joerg Arndt, Mar 26 2014

A203809 G.f.: exp( Sum_{n>=1} A000204(n)^9 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 9842, 97223, 58608265, 1390114224, 296390076414, 12122505505998, 1486321234837932, 84428445979241330, 7833461016478812734, 528228569507280147664, 43275470600883540869733, 3148637876123977595284117, 245565185017744596492591850

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 + ...
where
log(A(x)) = x + 3^9*x^2/2 + 4^9*x^3/3 + 7^9*x^4/4 + 11^9*x^5/5 + 18^9*x^6/6 + 29^9*x^7/7 + 47^9*x^8/8 + ... + Lucas(n)^9*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..530
Index entries for linear recurrences with constant coefficients, order 512.

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203808.

Cf. A203859, A203800.

CoefficientList[Series[1/((1 - x - x^2)^126*(1 + 4*x - x^2)^84*(1 - 11*x - x^2)^36*(1 + 29*x - x^2)^9*(1 - 76*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^9*x^k/k)+x*O(x^n)), n)}

{a(n,m=4)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

G.f.: 1/( (1-x-x^2)^126 * (1+4*x-x^2)^84 * (1-11*x-x^2)^36 * (1+29*x-x^2)^9 * (1-76*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203859(n) where A203859(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^8.

A203850 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-x^2)^n) / (1 + Lucas(n)x^n + (-x^2)^n) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -2, -4, 0, 14, 16, 0, 0, 4, -152, -188, 0, 0, -44, 0, 0, 4414, 5456, -4, 0, 1288, 0, 0, 0, 0, -335406, -414728, 0, 0, -97904, 0, 0, 4, 0, -8828, 0, 66770564, 82532956, 0, 0, 19483388, -304, 0, 0, 0, 1756816, 0, 0, 0, -34787592002, -42999828492, 0, 60508, -10150882544, 0, 0, 0, 0, -915304508, 0, 0, 796

Offset: 0

Paul D. Hanna, Jan 07 2012

sign

Compare to: Product_{n>=1} (1-q^n)/(1+q^n) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2), the Jacobi theta_4 function, which has the g.f: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * q^n/n ).

			G.f.: A(x) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...
-log(A(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 +...+ (sigma(2*n)-sigma(n))*Lucas(n)*x^n/n +...
Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2):
-log(H(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 8*x^4/4 + 12*x^5/5 + 16*x^6/6 + 16*x^7/7 +...+ (sigma(2*n)-sigma(n))*x^n/n +...
The g.f. equals the product:
A(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) *...
Positions of zeros form A022544:
[3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]
which are numbers that are not the sum of 2 squares.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A203860, A203861, A022544, A000204 (Lucas), A203318, A225524.

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

{a(n)=polcoeff(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)}

{a(n)=polcoeff(exp(sum(k=1, n,-(sigma(2*k)-sigma(k))*Lucas(k)*x^k/k)+x*O(x^n)), n)}

a(n) = 0 iff n is not the sum of 2 squares (A022544).
G.f.: Product_{n>=1} (1 - Lucas(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - Lucas(2*n)*x^(2*n) + x^(4*n)).
G.f.: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * Lucas(n) * x^n/n ) where Lucas(n) = A000204(n).

cp's OEIS Frontend

A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

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A156216 G.f.: A(x) = exp( Sum_{n>=1} A000204(n)^n * x^n/n ), a power series in x with integer coefficients.

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A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

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A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.

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A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

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A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).

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A203850 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-x^2)^n) / (1 + Lucas(n)x^n + (-x^2)^n) where Lucas(n) = A000204(n).