A000032 -id:A000032 - cp's OEIS frontend

A135407 Partial products of A000032 (Lucas numbers beginning at 2).

2, 2, 6, 24, 168, 1848, 33264, 964656, 45338832, 3445751232, 423827401536, 84341652905664, 27158012235623808, 14149324374760003968, 11927880447922683345024, 16269628930966540082612736

Offset: 0

Jonathan Vos Post, Dec 09 2007

This is to A000032 as A003266 is to A000045. a(n) is asymptotic to C*phi^(n*(n+1)/2) where phi=(1+sqrt(5))/2 is the golden ratio and C = 1.3578784076121057013874397... (see A218490). - Corrected and extended by Vaclav Kotesovec, Oct 30 2012

			a(0) = L(0) = 2.
a(1) = L(0)*L(1) = 2*1 = 2.
a(2) = L(0)*L(1)*L(2) = 2*1*3 = 6.
a(3) = L(0)*L(1)*L(2)*L(3) = 2*1*3*4 = 24.

G. C. Greubel, Table of n, a(n) for n = 0..95

Cf. A000032, A000045, A000204, A003266, A070825, A218490.

Rest[FoldList[Times,1,LucasL[Range[0,20]]]] (* Harvey P. Dale, Aug 21 2013 *)
Table[Round[GoldenRatio^(n(n+1)/2) QPochhammer[-1, GoldenRatio-2, n+1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 14 2016 *)

a(n) = prod(k=0, n, fibonacci(k+1)+fibonacci(k-1)); \\ Michel Marcus, Oct 13 2016

a(n) = Product_{k=0..n} A000032(k).

C = exp( Sum_{k>=1} 1/(k*(((3-sqrt(5))/2)^k-(-1)^k)) ). - Vaclav Kotesovec, Jun 08 2013

A247940 Least integer m > n such that m + n divides L(m) + L(n), where L(k) refers to the Lucas number A000032(k).

Original entry on oeis.org

5, 5, 15, 5, 19, 30, 17, 19, 15, 13, 13, 24, 35, 236, 33, 34, 31, 90, 29, 23, 27, 25, 25, 84, 47, 80, 45, 190, 43, 54, 41, 35, 39, 1216, 37, 72, 59, 212, 57, 43, 55, 66, 53, 86, 51, 76, 49, 60, 71, 53, 69, 55, 67, 222, 65, 122, 63, 112, 61, 264

Offset: 1

Zhi-Wei Sun, Sep 27 2014

nonn

Conjecture: Let A be any integer not congruent to 3 modulo 6. Define v(0) = 2, v(1) = A, and v(n+1) = A*v(n) + v(n-1) for n > 0. Then, for any integer n > 0, there are infinitely many positive integers m such that m + n divides v(m) + v(n).

This implies that a(n) exists for any n > 0.

			 a(3) = 15 since 15 + 3 = 18 divides L(15) + L(3) = 1364 + 4 = 18*76.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000032, A247824, A247937.

Do[m=n+1;Label[aa];If[Mod[LucasL[m]+LucasL[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

Original entry on oeis.org

0, 0, 0, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Inverse of the Lucas sequence (A000032), nearly, since a(Lucas(n))=n except for n=1 (see A130241 and A130247 for other versions). For n>=2, a(n+1) is equal to the partial sum of the Lucas indicator sequence (see A102460).

			a(10)=5, since Lucas(5)=11>=10 but Lucas(4)=7<10.

G. C. Greubel, Table of n, a(n) for n = 0..5000

For partial sums see A130244.
Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260.
Indicator sequence A102460.
Fibonacci inverse see A130233 - A130240, A104162.

Join[{0, 0, 0}, Table[Ceiling[Log[GoldenRatio, n + 1/2]], {n, 2, 50}]] (* G. C. Greubel, Dec 24 2017 *)

from itertools import islice, count
def A130242_gen(): # generator of terms
    yield from (0,0,0,2)
    a, b = 3, 4
    for i in count(3):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130242_list = list(islice(A130242_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = ceiling(log_phi((n+sqrt(n^2-4))/2))=ceiling(arccosh(n/2)/log(phi)) where phi=(1+sqrt(5))/2.
a(n) = A130241(n-1) + 1 = A130245(n-1) for n>=3.
G.f.: x/(1-x)*(2x^2+sum{k>=2, x^Lucas(k)}).
a(n) = ceiling(log_phi(n-1/2)) for n>=3, where phi is the golden ratio.

A005371 a(n) = L(L(n)), where L(n) are Lucas numbers A000032.

Original entry on oeis.org

3, 1, 4, 7, 29, 199, 5778, 1149851, 6643838879, 7639424778862807, 50755107359004694554823204, 387739824812222466915538827541705412334749, 19679776435706023589554719270187913247121278789615838446937339578603

Offset: 0

N. J. A. Sloane

T. Koshy (2001), Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 511-516
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..15

Cf. A000032, A005372, A007570, A068096, A068098.

[ Lucas(Lucas(n)): n in [0..20]]; // Vincenzo Librandi, Apr 16 2011

L:= n-> (<<0|1>, <1|1>>^n. <<2,1>>)[1,1]:
a:= n-> L(L(n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Jun 01 2016

l[n_]:= l[n]= l[n-1] + l[n-2]; l[0]= 2; l[1]= 1; Table[l[l[n]], {n,0,12}]
LucasL[LucasL[Range[0, 15]]] (* G. C. Greubel, Dec 21 2017 *)

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,15, print1(lucas(lucas(n)), ", ")) \\ G. C. Greubel, Dec 21 2017

[lucas_number2(lucas_number2(n, 1,-1),1,-1) for n in range(15)] # G. C. Greubel Nov 14 2022

More terms from Mario Catalani (mario.catalani(AT)unito.it), Mar 14 2003

Offset changed Feb 28 2007

A205114 Least k such that n divides L(k)-L(j) for some j satisfying 1<=jA000032).

Original entry on oeis.org

2, 2, 3, 4, 5, 4, 5, 5, 7, 5, 6, 8, 7, 6, 6, 10, 6, 7, 10, 8, 10, 7, 8, 9, 7, 7, 16, 7, 8, 10, 16, 11, 11, 11, 10, 8, 13, 10, 11, 8, 11, 14, 8, 8, 12, 8, 9, 11, 11, 16, 13, 13, 12, 16, 12, 10, 17, 9, 14, 10

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

See A204892 for a discussion and guide to related sequences.

Cf. A000032, A204892, A205119.

s[n_] := s[n] = LucasL[n]; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]    (* A000032 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]    (* A205112 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]     (* A205113 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A205114 *)
Table[j[n], {n, 1, z2}]       (* A205115 *)
Table[s[k[n]], {n, 1, z2}]    (* A205116 *)
Table[s[j[n]], {n, 1, z2}]    (* A205117 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205118 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205119 *)

A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 4, 1, 2, 2, 4, 2, 3, 1, 3, 3, 2, 2, 5, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 2, 4, 2, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 5, 1, 2, 2, 4, 3, 3, 1, 3, 2, 2, 1, 5, 1, 2, 3, 4, 1, 4, 2, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000032, A093540, A102460, A304091, A304094, A304096.

Cf. also A005086, A300837.

Module[{nn=11,luc},luc=LucasL[Range[0,nn]];Table[Count[n/luc,?IntegerQ],{n,Max[luc]}]] (* _Harvey P. Dale, Jul 01 2023 *)

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304092(n) = sumdiv(n,d,A102460(d));

a(n) = Sum_{d|n} A102460(d).
a(n) = A304091(n) + A102460(n).
a(n) = A304094(n) + A059841(n) = A304096(n) + A059841(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 + 1/2 = 2.462858... . - Amiram Eldar, Dec 31 2023

A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 49, 121, 169, 289, 361, 529, 841, 961, 1127, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 3751, 4181, 4489, 4901, 4961, 5041, 5329, 5777, 6241, 6721, 6889, 7381, 7921, 9409, 10201, 10609, 10877, 11449, 11881, 12769, 13201, 15251, 16129, 17161, 18081, 18769, 19321

Offset: 1

Ovidiu Bagdasar, Nov 24 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity
V(p-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).
For a=1 and b=-1, we have D=5 and V(m) recovers A000032(m) (Lucas numbers).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

Cf. A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 5])] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

Original entry on oeis.org

3, 6, 24, 120, 960, 11520, 218880, 6566400, 315187200, 24269414400, 3009407385600, 601881477120000, 194407717109760000, 101480828331294720000, 85649819111612743680000, 116912003087351395123200000, 258141702816871880432025600000

Offset: 0

Clark Kimberling, Jul 25 2024

nonn

Cf. A000032, A000142, A003266, A070825, A082480, A135407, A374655.

w[n_] := Product[LucasL[k] + 1, {k, 0, n}]
Table[w[n], {n, 0, 20}]

a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

A057854 Non-Lucas numbers: the complement of A000032.

Original entry on oeis.org

5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80

Offset: 1

Roger Cuculière, Nov 12 2000

The formula is a consequence of the Lambek-Moser theorem.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Lambek-Moser theorem

a := proc(n) floor(-1/ln(1/2+1/2*5^(1/2))*LambertW(-1,-ln(1/2+1/2*5^(1/2))/ ((1/2+1/2*5^(1/2))^(n+1/2)))+1/2) end; # Simon Plouffe, Nov 30 2017
# alternative
isA000032 := proc(n)
    local l1,l2 ;
    if n <= 0 then
        false;
    elif n <= 4 then
        true ;
    else
        l1 := 3 ; l2 := 4 ;
        while true do
            l := l1+l2 ;
            if l > n then
                return false;
            elif l = n then
                return true;
            else
                l1 := l2 ; l2 := l ;
            end if;
        end do:
    end if;
end proc:
isA057854 := proc(n)
    not isA000032(n) ;
end proc:
A057854 := proc(n)
    option remember;
    if n = 1 then
        5 ;
    else
        for a from procname(n-1)+1 do
            if isA057854(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A057854(n),n=1..10) ; # R. J. Mathar, Feb 01 2019

a[n_] := With[{phi = (1 + Sqrt[5])/2}, Floor[1/2 - LambertW[-1, -Log[phi]/phi^(n + 1/2)]/Log[phi]]];
Table[a[n], {n, 1, 70}] (* Peter Luschny, Nov 30 2017 *)
b:= Complement[Range[1, 100], LucasL[Range[20]]]; Table[b[[n+1]], {n, 1, 70}] (* G. C. Greubel, Jun 19 2019 *)

def A057854(n):
    def f(x):
        if x<=2: return n+2
        a, b, c = 1, 3, 0
        while b<=x:
            a, b = b, a+b
            c += 1
        return n+c+2
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

a(n) = floor(1/2 - LambertW(-1, -log(phi)/phi^(n+1/2))/log(phi)) with phi = (1+sqrt(5))/2. - Nicolas Normand (nicolas.normand (at) polytech.univ-nantes.fr)

a(n) = A090946(n+2). - R. J. Mathar, Jan 29 2019

More terms from Larry Reeves (larryr(AT)acm.org), Nov 28 2000

A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 9, 39, 120, 315, 753, 1687, 3612, 7470, 15040, 29634, 57366, 109421, 206115, 384105, 709152, 1298613, 2360943, 4264835, 7659870, 13686456, 24340184, 43102644, 76031100, 133636825, 234116493, 408900987

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A004799, A060922.

I:=[1,9,39,120,315,753]; [n le 6 select I[n] else 3*Self(n-1) - 5*Self(n-3) + 3*Self(n-5) + Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[((1 + 2*x)/(1 - x - x^2))^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1}, {1,9,39,120,315,753}, 30] (* G. C. Greubel, Dec 21 2017 *)

x='x+O('x^30); Vec(((1+2*x)/(1-x-x^2))^3) \\ G. C. Greubel, Dec 21 2017

G.f.: ((1+2*x)/(1-x-x^2))^3.
a(n) = A060922(n+2, 2) (third column of Lucas triangle).
a(n) = (n+1)*((5*n+4)*L(n+2) + (5*n-2)*L(n+1))/10, n >= 1, with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.

cp's OEIS Frontend

A135407 Partial products of A000032 (Lucas numbers beginning at 2).

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A247940 Least integer m > n such that m + n divides L(m) + L(n), where L(k) refers to the Lucas number A000032(k).

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A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

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A005371 a(n) = L(L(n)), where L(n) are Lucas numbers A000032.

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A205114 Least k such that n divides L(k)-L(j) for some j satisfying 1<=jA000032).

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A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

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A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

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