A000204 -id:A000204 - cp's OEIS frontend

A203854 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^3, where Lucas(n) = A000204(n).

1, 13, 21, 79, 266, 957, 3484, 12935, 48768, 185951, 716418, 2781675, 10878520, 42789478, 169181010, 671866245, 2678678730, 10716651456, 43007270292, 173072549610, 698235680844, 2823329210391, 11439823946306, 46440709210035, 188856966693230, 769241291729020

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^13 * (1-4*x^3-x^6)^21 * (1-7*x^4+x^8)^79 * (1-11*x^5-x^10)^266 * (1-18*x^6+x^12)^957 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^4 * x^n/n ) = g.f. of A203804:
F(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
log(F(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 1..360

Cf. A203804, A203853, A203855, A203856, A203857, A203858, A203859, A203800.

a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^3&]; Array[a, 30] (* Jean-François Alcover, Dec 04 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^3)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^4*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)^4 * x^n/n), which is the g.f. of A203804.

a(n) ~ phi^(3*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203855 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^4, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 40, 85, 580, 2928, 17440, 101040, 609660, 3706880, 22887192, 142567200, 895855380, 5667708960, 36072949560, 230763023408, 1482822818820, 9565561745040, 61920953016320, 402074969960400, 2618069854211784, 17090016552803440, 111812320834030800

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^40 * (1-4*x^3-x^6)^85 *
(1-7*x^4+x^8)^580 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^17440 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^5 * x^n/n ) = g.f. of A203805:
F(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 +...
where
log(F(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 +...

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A203805, A203853, A203854, A203856, A203857, A203858, A203859, A203800.

a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^4 &]; Array[a, 30] (* Jean-François Alcover, Dec 04 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^4)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^5*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)^5 * x^n/n), which is the g.f. of A203805.

a(n) ~ phi^(4*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203856 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^5, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 121, 341, 4141, 32210, 314717, 2930164, 28666025, 281724928, 2815289555, 28370872818, 288468152625, 2952876368200, 30409537607218, 314760765272250, 3272590619892675, 34158620991538050, 357779277130203136, 3758998894159780092, 39603542856374168550

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^121 * (1-4*x^3-x^6)^341 * (1-7*x^4+x^8)^4141 * (1-11*x^5-x^10)^32210 * (1-18*x^6+x^12)^314717 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^6 * x^n/n ) = g.f. of A203806:
F(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
where
log(F(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 = 1..955

Cf. A203806, A203853, A203854, A203855, A203857, A203858, A203859, A203800.

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^5&]/n; Array[a, 20] (* Jean-François Alcover, Dec 07 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^5)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^6*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)^6 * x^n/n), which is the g.f. of A203806.

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

A203857 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 364, 1365, 29230, 354312, 5667900, 84974760, 1347387210, 21411102720, 346282421940, 5645803690800, 92886793449030, 1538448587832240, 25635241395476100, 429333683845968552, 7222607529064709670, 121980435560782376760, 2067248664062116147200

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^364 * (1-4*x^3-x^6)^1365 * (1-7*x^4+x^8)^29230 * (1-11*x^5-x^10)^354312 * (1-18*x^6+x^12)^5667900 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^7 * x^n/n ) = g.f. of A203807:
F(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where log(F(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 = 1..795

Cf. A203807, A203853, A203854, A203855, A203856, A203858, A203859, A203800.

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^6&]/n; Array[a, 18] (* Jean-François Alcover, Dec 07 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^6)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^7*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)^7 * x^n/n), which is the g.f. of A203807.

a(n) ~ phi^(6*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203858 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^7, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 1093, 5461, 205339, 3897434, 102033577, 2464268044, 63327787115, 1627243839488, 42592757748839, 1123514934469218, 29909548355299575, 801531714260597080, 21610508530971123358, 585611144766061271010, 15940294818101008311105, 435592135387553867410170

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^1093 * (1-4*x^3-x^6)^5461 * (1-7*x^4+x^8)^205339 * (1-11*x^5-x^10)^3897434 * (1-18*x^6+x^12)^102033577 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^8 * x^n/n ) = g.f. of A203808:
F(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(F(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 = 1..500

Cf. A203808, A203853, A203854, A203855, A203856, A203857, A203859, A203800.

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^7 &]/n; Array[a, 50] (* G. C. Greubel, Mar 05 2018 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^7)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^8*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)^8 * x^n/n), which is the g.f. of A203808.

a(n) ~ phi^(7*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203859 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^8, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 3280, 21845, 1439560, 42871776, 1836648080, 71463773280, 2976410112120, 123670531932160, 5238909421389744, 223579471959374400, 9630874585937597160, 417598023129771078720, 18217658692611614215920, 798773601460909332885856, 35180230663617319463871240

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^3280 * (1-4*x^3-x^6)^21845 * (1-7*x^4+x^8)^1439560 * (1-11*x^5-x^10)^42871776 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^9 * x^n/n ) = g.f. of A203809:
F(x) = 1 + x + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 +...
where
log(F(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 +...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A203809, A203853, A203854, A203855, A203856, A203857, A203858, A203800.

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^8&]/n; Array[a, 30] (* G. C. Greubel, Dec 19 2017 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^8)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^9*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)^9 * x^n/n), which is the g.f. of A203809.

a(n) ~ phi^(8*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A153173 a(n) = L(5*n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

11, 41, 341, 2161, 15251, 103361, 711491, 4868641, 33391061, 228811001, 1568437211, 10749853441, 73681573691, 505018447961, 3461454668501, 23725145626561, 162614613425891, 1114577020834241, 7639424866266611

Offset: 1

Artur Jasinski, Dec 20 2008

All numbers in this sequence are congruent to 1 mod 10.

G. C. Greubel, Table of n, a(n) for n = 1..1000
L. Carlitz, Problem B-185, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 325; Lucas Ratio I, Solution to Problem B-185 by C. B. A. Peck, ibid., Vol. 9, No. 1 (1971), p. 109.
L. Carlitz, Problem B-186, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 326; Lucas Ratio II, Solution to Problem B-186 by John Wessner, ibid., Vol. 9, No. 1 (1971), pp. 109-110.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000032, A000045, A000204, A005248, A056854, A085695, A110391.

I:=[11, 41, 341, 2161, 15251]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[5*n]/LucasL[n], {n, 1, 50}]
CoefficientList[Series[x*(11-14*x-29*x^2+6*x^3+x^4)/((1-x)*(x^2-7*x+1)*(x^2+3*x+1)), {x,0,50}], x] (* G. C. Greubel, Dec 21 2017 *)
a[ n_] := 1 + 5*Fibonacci[n]*Fibonacci[3*n]; (* Michael Somos, Apr 23 2022 *)

{L(n)=fibonacci(n-1)+fibonacci(n+1)}; a(n) = L(5*n)/L(n) \\ Charles R Greathouse IV, Jun 11 2015

my(x='x+O('x^30)); Vec(x*(11-14*x-29*x^2+6*x^3+x^4 )/((1-x)*(x^2-7*x +1)*(x^2+3*x+1))) \\ G. C. Greubel, Dec 21 2017

{a(n) = 1 + 5*fibonacci(n)*fibonacci(3*n)}; /* Michael Somos, Apr 23 2022 */

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: -x*(11-14*x-29*x^2+6*x^3+x^4)/( (x-1)*(x^2-7*x+1)*(x^2+3*x+1) ).
a(n) = 1 + A056854(n) - (-1)^n*A005248(n). (End)
From Amiram Eldar, Feb 02 2022: (Start)
a(n) = Lucas(2*n)^2 - (-1)^n*Lucas(2*n) - 1 (Carlitz, Problem B-185).
a(n) = (Lucas(2*n) - 3*(-1)^n)^2 + (-1)^n*(5*Fibonacci(n))^2 (Carlitz, Problem B-186). (End)
a(n) = a(-n) = 1 + 10*A085695(n) = 5 + L(n-1)*L(n)^2*L(n+1) for all n in Z. - Michael Somos, Apr 23 2022

A246453 Lucas numbers (A000204) of the form n^2 + 2.

Original entry on oeis.org

3, 11, 18, 123, 843, 5778, 39603, 271443, 1860498, 12752043, 87403803, 599074578, 4106118243, 28143753123, 192900153618, 1322157322203, 9062201101803, 62113250390418, 425730551631123, 2918000611027443, 20000273725560978, 137083915467899403, 939587134549734843

Offset: 1

Michel Lagneau, Aug 26 2014

a(n) = {11} union {A000204(2+4*n)} for n=0,1,...

Intersection of A000204 and A059100. - Michel Marcus, Aug 26 2014

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-1).

Cf. A000204 (Lucas), A059100 (n^2+2).

Cf. quadrisection of A000032: A056854 (first), A056914 (second), this sequence (third, without 11), A288913 (fourth).

I:=[3,11,18,123]; [n le 4 select I[n] else 7*Self(n-1)-Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

with(combinat,fibonacci):lst:={}:lst1:={}:nn:=5000:
  for n from 1 to nn do:
    lst:=lst union {2*fibonacci(n-1)+fibonacci(n)}:
  od:
   for m from 1 to nn do:
    if {m^2+2} intersect lst = {m^2+2}
    then
    lst1:=lst1 union {m^2+2}:
    else
    fi:
   od:
   print(lst1):

CoefficientList[Series[x*(3-10*x-56*x^2+8*x^3)/(1-7*x+x^2), {x,0,50}], x] (* or *) LinearRecurrence[{7,-1}, {3, 11, 18, 123}, 30] (* G. C. Greubel, Dec 21 2017 *)
Select[LucasL[Range[100]],IntegerQ[Sqrt[#-2]]&] (* Harvey P. Dale, Dec 31 2018 *)

lista(nn) = for (n=0, nn, luc = fibonacci(n+1) + fibonacci(n-1); if (issquare(luc-2), print1(luc, ", "))); \\ Michel Marcus, Mar 29 2016

Vec(x*(3 - 10*x - 56*x^2 + 8*x^3) / (1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017

From Colin Barker, Jun 20 2017: (Start)
G.f.: x*(3 - 10*x - 56*x^2 + 8*x^3) / (1 - 7*x + x^2).
a(n) = (2^(-n)*((7+3*sqrt(5))^n*(-20+9*sqrt(5)) + (7-3*sqrt(5))^n*(20+9*sqrt(5)))) / sqrt(5) for n>2.
a(n) = 7*a(n-1) - a(n-2) for n>4. (End)
E.g.f.: 2*exp(7*x/2)*(9*cosh(3*sqrt(5)*x/2) - 4*sqrt(5)*sinh(3*sqrt(5)*x/2)) + 4*x^2 - 18. - Stefano Spezia, Apr 14 2025

Corrected by Michel Marcus, Mar 29 2016

A086600 Number of primitive prime factors in the n-th Lucas number A000204(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 3, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 2

Offset: 1

T. D. Noe, Jul 24 2003

A prime factor of Lucas(n) is called primitive if it does not divide Lucas(r) for any r < n. It can be shown that there is at least one primitive prime factor for n > 6. When n is prime, all the prime factors of Lucas(n) are primitive.

			a(22) = 2 because Lucas(22) = 3*43*307 and neither 43 nor 307 divide a smaller Lucas number.

Max Alekseyev, Table of n, a(n) for n = 1..1411 (first 1000 terms from T. D. Noe)
Blair Kelly, Fibonacci and Lucas Factorizations,
Eric Weisstein's World of Mathematics, Lucas Number.

Cf. A000204 (Lucas numbers), A058036, A086598 (number of distinct prime factors), A086599 (number of prime factors, counting multiplicity), A274333.

lst:=[]; pr:=1; for n in [1..105] do pd:=PrimeDivisors(Lucas(n)); d:=1; t:=0; for c in [1..#pd] do f:=pd[c]; if Gcd(pr, f) eq 1 then t+:=1; else d:=d*f; end if; end for; Append(~lst, t); pr:=pr*Truncate(Lucas(n)/d); end for; lst; // Arkadiusz Wesolowski, Jun 22 2016

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; pLst={}; Join[{0}, Table[f=Transpose[FactorInteger[Lucas[n]]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 2, 150}]]

a(n) = Sum{d|n and n/d odd} mu(n/d) a(d) -1 if 6|n and n/6 is a power of 2.

A156234 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)A000204(n)x^n/n ).

Original entry on oeis.org

1, 1, 5, 10, 30, 63, 170, 355, 880, 1875, 4349, 9189, 20810, 43355, 95140, 198247, 424527, 875965, 1849535, 3781820, 7873167, 16005196, 32883560, 66390850, 135198990, 271051271, 546931398, 1090751095, 2183512495, 4329540830

Offset: 0

Paul D. Hanna, Feb 06 2009

nonn

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),

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

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 + ...
log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + ...
Also, the g.f. equals the product:
A(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) * ...).

Robert Israel, Table of n, a(n) for n = 0..3000

Cf. A225528, A000203 (sigma), A000204 (Lucas), A000041 (partitions), A000045.

N:= 100: # to get a(0) to a(N)
G:= exp(add(numtheory:-sigma(n)*lucas(n)*x^n/n,n=1..N)):
S:= series(G,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Dec 23 2015

{a(n)=polcoeff(exp(sum(k=1,n,sigma(k)*(fibonacci(k-1)+fibonacci(k+1))*x^k/k)+x*O(x^n)),n)}
for(n=0,40,print1(a(n),", "))

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(prod(m=1,n,1/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

a(n) = (1/n)*Sum_{k=1..n} sigma(n)*A000204(k)*a(n-k) for n>0, with a(0) = 1.
G.f.: Product_{n>=1} 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).
Logarithmic derivative yields A225528.

cp's OEIS Frontend

A203854 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^3, where Lucas(n) = A000204(n).

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A203855 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^4, where Lucas(n) = A000204(n).

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A203856 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^5, where Lucas(n) = A000204(n).

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A203857 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).

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A203858 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^7, where Lucas(n) = A000204(n).

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A203859 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^8, where Lucas(n) = A000204(n).

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A153173 a(n) = L(5*n)/L(n) where L(n) = Lucas number A000204(n).

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A156234 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)A000204(n)x^n/n ).