A000204 -id:A000204 - cp's OEIS frontend

A203861 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n))^3 where Lucas(n) = A000204(n).

1, -3, -9, 20, 45, 0, -151, -231, 0, 140, 1107, 2052, 49, -1305, 0, -15004, -28260, 0, 17710, 0, 81, 324040, 589953, 0, -375570, -1089, 0, -124124, -10659705, -19764180, -121, 12605358, 0, 0, 4158315, 0, 567552368, 1052295189, -780030, -669901660, 0, 0, -221399431, -85965, 0

Offset: 0

Paul D. Hanna, Jan 07 2012

sign

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

			G.f.: A(x) = 1 - 3*x - 9*x^2 + 20*x^3 + 45*x^4 - 151*x^6 - 231*x^7 +...
-log(A(x))/3 = 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 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2)^3 * (1-3*x^2+x^4)^3 * (1-4*x^3-x^6)^3 * (1-7*x^4+x^8)^3 * (1-11*x^5-x^10)^3 * (1-18*x^6+x^12)^3 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...]
which are numbers that are not the sum of two triangular numbers.

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

Cf. A203860, A203850, A156234, A020757.

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

{a(n)=polcoeff(exp(sum(k=1, n, -3*sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}

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

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A000204(n) * x^n/n ).

A022801 n-th Lucas number (A000204(n)) + n-th non-Lucas number (A090946(n+1)).

Original entry on oeis.org

3, 8, 10, 15, 20, 28, 41, 60, 90, 138, 215, 339, 540, 863, 1385, 2229, 3594, 5802, 9374, 15153, 24503, 39631, 64109, 103713, 167793, 271476, 439238, 710682, 1149887, 1860535, 3010387, 4870886, 7881236, 12752084, 20633281, 33385325, 54018565

Offset: 1

Clark Kimberling

with(combinat): lucas:= n->fibonacci(n+1)+fibonacci(n-1): n:=1: for k from 1 to 7 do for nonlucas from lucas(k)+1 to lucas(k+1)-1 do printf("%d, ",nonlucas+lucas(n)) :n:=n+1 od od: # C. Ronaldo (aga_new_ac(AT)hotmail.com)

Module[{nn=40,luc,nluc},luc=LucasL[Range[nn]];nluc=Complement[Range[ Last[ luc]],luc];Total/@Thread[{luc,Take[nluc,Length[luc]]}]] (* Harvey P. Dale, May 02 2019 *)

from sympy import lucas
def A022801(n):
    def f(x):
        if x<=2: return n+1
        a, b, c = 1, 3, 0
        while b<=x:
            a, b = b, a+b
            c += 1
        return n+1+c
    m, k = n+1, f(n+1)
    while m != k: m, k = k, f(k)
    return m+lucas(n) # Chai Wah Wu, Sep 10 2024

a(n) = A000204(n) + A090946(n + 1). - Sean A. Irvine, May 21 2019

Thanks to Karima MOUSSAOUI (bouyao(AT)wanadoo.fr), who noticed that there were two versions of this sequence, differing at about the 22nd term, Feb 28 2004

More terms from Emeric Deutsch, Jan 14 2005

A065156 Numbers n such that some Lucas number (A000204) is divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 19, 22, 23, 27, 29, 31, 38, 41, 43, 44, 46, 47, 49, 54, 58, 59, 62, 67, 71, 76, 79, 81, 82, 83, 86, 94, 98, 101, 103, 107, 116, 118, 121, 123, 124, 127, 129, 131, 134, 139, 142, 151, 158, 161, 162, 163, 166, 167, 179, 181, 191, 199

Offset: 1

Dean Hickerson, Oct 18 2001

From A.H.M. Smeets, Sep 20 2020 (Start)
For the Fibonacci numbers, each natural number divides some Fibonacci number (see A001177).
If, for some number m, m divides some Lucas number L_i (=A000204(i)), then, the smallest i satisfies i <= m. (End)

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe)
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5

Complement of A064362. Cf. A000204.

Cf. A001177, A053032, A140409.

test[ n_ ] := For[ a=1; b=3, True, t=b; b=Mod[ a+b, n ]; a=t, If[ b==0, Return[ True ] ]; If[ a==2&&b==1, Return[ False ] ] ]; Select[ Range[ 200 ], test ]
Take[Flatten[Divisors/@LucasL[Range[200]]]//Union,70] (* Harvey P. Dale, Jun 07 2020 *)

a, n = 0, 0
while n < 1000:
    a, f0, f1, i = a+1, 1, 2, 1
    if f1%a == 0:
        n = n+1
        print(n,a)
    else:
        while f0%a != 0 and i <= a:
            f0, f1, i = f0+f1, f0, i+1
        if i <= a:
            n = n+1
            print(n,a) # A.H.M. Smeets, Sep 20 2020

Equals {1,2,4} union {p^e | p in A140409 and e > 0} union {2*p^e | p in A140409 and e > 0} union {4*p | p in A053032} union {4*p*q | p, q in A053032}. - A.H.M. Smeets, Sep 20 2020

A130774 Cumulative concatenation of A000204 Lucas numbers (beginning at 1).

Original entry on oeis.org

1, 13, 134, 1347, 134711, 13471118, 1347111829, 134711182947, 13471118294776, 13471118294776123, 13471118294776123199, 13471118294776123199322, 13471118294776123199322521, 13471118294776123199322521843

Offset: 1

Jonathan Vos Post, Aug 19 2007

a(2) = 13 and a(10) = 13471118294776123 are prime. What is the next prime?

The next prime is a(31), a number of 120 digits. - Giovanni Resta, Jun 18 2016

Cf. A000204.

Module[{nn=20,ll},ll=LucasL[Range[nn]];Table[FromDigits[Flatten[IntegerDigits/@Take[ll,n]]],{n,nn}]] (* Harvey P. Dale, May 07 2023 *)

a(1) = 1; a(n+1) = Concatenate(a(n),A000204(n+1)).

A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 149, 151, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 307, 337, 347, 349, 379, 397, 401, 419, 421, 449, 463, 487, 523, 541, 571, 643, 647, 661

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: The sequence has infinitely many terms.

This follows from the conjecture in A233359.

			a(1) = 2 since L(1) + q(1) = 1 + 1 = 2.
a(2) = 3 since L(1) + q(3) = 1 + 2 = 3.
a(3) = 5 since L(2) + q(3) = 3 + 2 = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..270

Cf. A000009, A000040, A000032, A000204, A232504, A233307, A233346, A233359.

n=0
Do[Do[If[LucasL[j]>=Prime[m],Goto[aa],
Do[If[PartitionsQ[k]==Prime[m]-LucasL[j],
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]>Prime[m]-LucasL[j],Goto[bb]];Continue,{k,1,2*(Prime[m]-LucasL[j])}]];
Label[bb];Continue,{j,1,2*Log[2,Prime[m]]}];
Label[aa];Continue,{m,1,125}]

A288039 Number of compositions (ordered partitions) of n into Lucas numbers (beginning with 1) (A000204).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 16, 27, 43, 70, 118, 195, 318, 524, 868, 1430, 2351, 3878, 6399, 10542, 17367, 28634, 47206, 77793, 128212, 211346, 348360, 574153, 946342, 1559849, 2571016, 4237616, 6984659, 11512526, 18975464, 31276187, 51550993, 84968944, 140049801, 230836734, 380476447, 627119783, 1033648857

Offset: 0

Ilya Gutkovskiy, Jun 04 2017

nonn

			a(4) = 4 because we have [4], [3, 1], [1, 3] and [1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for sequences related to compositions

Cf. A000204, A003263, A067592, A076739, A080888.

CoefficientList[Series[1/(1 - Sum[x^LucasL[k], {k, 1, 15}]), {x, 0, 43}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A000204(k)).

A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 1, 2, 2, 1, 3, 1, 3, 2, 1, 1, 3, 2, 1, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 3, 2, 1, 2, 2, 1, 1, 4, 1, 1, 3, 3, 1, 3, 2, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000204, A093540, A304091, A304094, A304095.

Cf. also A293435, A300836.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304093(n) = sumdiv(n,d,(dA000204(d));

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540. - Amiram Eldar, Jul 05 2025

A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000204, A093540, A304092, A304093, A304096.

Cf. also A005086, A300837.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304094(n) = sumdiv(n,d,isA000204(d));

a(n) = A304092(n) - A059841(n).
a(n) = A304096(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 = 1.962858... . - Amiram Eldar, Dec 31 2023

A134072 Concatenation of A000204 Lucas numbers (beginning at 1) in reverse order.

Original entry on oeis.org

1, 31, 431, 7431, 117431, 18117431, 2918117431, 472918117431, 76472918117431, 12376472918117431, 19912376472918117431, 32219912376472918117431, 52132219912376472918117431, 84352132219912376472918117431

Offset: 1

Alexander Adamchuk, Oct 06 2007

Indices of prime terms are {2, 3, 5, 11, ...}. Primes are listed in A134071 = {31, 431, 117431, 19912376472918117431, ...}.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134070 (primes in A130774).
Cf. A134071 (primes in A134072).

Module[{nn=20,lnos},lnos=LucasL[Range[nn]];Table[FromDigits[Flatten[ IntegerDigits/@ Reverse[Take[lnos,n]]]],{n,nn}]] (* Harvey P. Dale, Jul 27 2015 *)

Edited by Charles R Greathouse IV, Apr 26 2010

A153180 a(n) = L(13n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

521, 90481, 35355581, 10525900321, 3489827263001, 1111126318086721, 359316586176453881, 115509240442846111681, 37216910406644366498621, 11980863523543017476802001, 3858153294795970321295258921

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 1 mod 10
congruent to 1 mod 100 (iff n is congruent to 0 mod 5),Q congruent to 21 mod 100 (iff n is congruent to 1 or 4 mod 5),
congruent to 81 mod 100 (iff n is congruent to 2 or 3 mod 5).Q

Index entries for linear recurrences with constant coefficients, signature (233, 33552, -1493064, -27372840, 186135312, 488605194, -488605194, -186135312, 27372840, 1493064, -33552, -233, 1).

A000032, A000204, A047221, A110391, A153173, A153175, A153177, A153179.

Table[LucasL[13 n]/LucasL[n], {n, 1, 150}]

a(n)= +233*a(n-1) +33552*a(n-2) -1493064*a(n-3) -27372840*a(n-4) +186135312*a(n-5) +488605194*a(n-6) -488605194*a(n-7) -186135312*a(n-8) +27372840*a(n-9) +1493064*a(n-10) -33552*a(n-11) -233*a(n-12) +a(n-13). G.f.: -1+ (-2-123*x)/(x^2+123*x+1) +(2-322*x)/(x^2-322*x+1) +(-2-3*x)/(x^2+3*x+1) +(2-7*x)/(x^2-7*x+1) +(2-47*x)/(x^2-47*x+1) -1/(x-1)+ (-2-18*x)/(x^2+18*x+1). [From R. J. Mathar, Oct 22 2010]

cp's OEIS Frontend

A203861 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 where Lucas(n) = A000204(n).

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A022801 n-th Lucas number (A000204(n)) + n-th non-Lucas number (A090946(n+1)).

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A065156 Numbers n such that some Lucas number (A000204) is divisible by n.

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A130774 Cumulative concatenation of A000204 Lucas numbers (beginning at 1).

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A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

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A288039 Number of compositions (ordered partitions) of n into Lucas numbers (beginning with 1) (A000204).

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A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

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A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

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A203861 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n))^3 where Lucas(n) = A000204(n).