A000204 -id:A000204 - cp's OEIS frontend

A166169 a(n) = Lucas(n^2) = A000204(n^2) for n >= 1.

1, 7, 76, 2207, 167761, 33385282, 17393796001, 23725150497407, 84722519070079276, 792070839848372253127, 19386725908489881939795601, 1242282009792667284144565908482, 208406472252232726621841472637412401

Offset: 1

Paul D. Hanna, Oct 08 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..35

Cf. A166168, A000204, A000045.

[ Lucas(n^2) : n in [1..50]]; // Vincenzo Librandi, Apr 14 2011

Table[LucasL[n^2], {n, 1, 10}] (* G. C. Greubel, May 06 2016 *)

PARI
```
a(n)=fibonacci(n^2-1)+fibonacci(n^2+1)
```

Logarithmic derivative of A166168.
a(n) = ((1 + sqrt(5))/2)^(n^2) + ((1 - sqrt(5))/2)^(n^2).
a(n) = Fibonacci(n^2 - 1) + Fibonacci(n^2 + 1).

A204291 G.f.: Sum_{n>=1} moebius(n)x^n/(1 - Lucas(n)x^n + (-1)^nx^(2n)), where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 0, 1, 0, 4, -3, 12, 0, 17, -10, 88, -54, 232, -28, 184, 0, 1596, -969, 4180, -1230, 4632, -198, 28656, -17388, 60020, -520, 98209, -23604, 514228, -461932, 1346268, 0, 1722688, -3570, 6672168, -5598882, 24157816, -9348, 31351552, -18606210, 165580140

Offset: 1

Paul D. Hanna, Jan 14 2012

sign

Compare g.f. to the identity: x = Sum_{n>=1} moebius(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

			G.f.: A(x) = x + x^3 + 4*x^5 - 3*x^6 + 12*x^7 + 17*x^9 - 10*x^10 + 88*x^11 +...
where A(x) = x/(1-x-x^2) - x^2/(1-3*x^2+x^4) - x^3/(1-4*x^3-x^6) - x^5/(1-11*x^5-x^10) + x^6/(1-18*x^6+x^12) +...+ moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Paul D. Hanna, Table of n, a(n) for n = 1..1024

Cf. A203847, A000204 (Lucas), A000045, A008683.

with(numtheory): seq(add(mobius(d)*combinat[fibonacci](n)/combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 09 2025

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

a(k) = 0 iff k = 2^n for n>=1.

a(n) = Fibonacci(n) * Sum_{d|n} mu(d)/Fibonacci(d). - Ridouane Oudra, Apr 09 2025

A225528 a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

Original entry on oeis.org

1, 9, 16, 49, 66, 216, 232, 705, 988, 2214, 2388, 9016, 7294, 20232, 32736, 68417, 64278, 225342, 186980, 635334, 783232, 1425708, 1537896, 6220920, 5200591, 11400606, 17568160, 39796232, 34495530, 133955856, 96331168, 306863361, 378297408, 688610322, 990395472, 3038060662

Offset: 1

Paul D. Hanna, May 09 2013

nonn

			L.g.f.: L(x) = x + 9*x^2/2 + 16*x^3/3 + 49*x^4/4 + 66*x^5/5 + 216*x^6/6 +...
which is equivalent to:
L(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 + 8*29*x^7/7 + 15*47*x^8/8 +...+ sigma(n)*Lucas(n)*x^n/n +...
where exponentiation yields the g.f. of A156234:
exp(L(x)) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 +...+ A156234(n)*x^n +...
and equals the product:
exp(L(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 + (-x^2)^n) *...).

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A156234, A225525.

{a(n)=sigma(n)*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1,40,print1(a(n),", "))

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

L.g.f.: Sum_{n>=1} -log(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} a(n)*x^n/n.

Logarithmic derivative of A156234.

A294203 Number of partitions of n into distinct Lucas parts (A000204) greater than 1.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 24 2017

nonn

Convolution of the sequences A003263 and A033999.
Positions of 0: 1, 2, 5, 6, 8, 9, 12, 13, ... = A287775(n) - 1 (conjecture).
From Michel Dekking, Dec 30 2017: (Start)
Proof of the 'positions of 0' conjecture: let (z(n))=1,2,5,6,8,9,12,... be the positions of 0. The crucial observation is that if a number n is the sum of distinct Lucas parts greater than 1, then n+1 is a sum of Lucas parts. This implies that (z(2n))=2,6,9,13,... is the sequence of numbers A054770 that are not a sum of Lucas numbers. We see there that Ian Agol proved that b(n):=A054770(n)=floor(phi*n)+2n-1. But then the sequence of first differences (b(n+1)-b(n)) equals the Fibonacci word on the alphabet {4,3}, yielding that (z(2n)-z(2n-1)) equals the Fibonacci word on {3,2}, and we already know that z(2n+1)-z(2n)=1 for all n. On the other hand, A287775 has the same first difference sequence given by A108103. Since A287775(1)=2, the conjecture follows. (End)
Positions of 1: 0, 3, 4, 10, 15, 28, 44, 75, ... = A001350(n+1) - 1 (conjecture).

			a(7) = 2 because we have [7] and [4, 3].

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences related to partitions

Cf. A000204, A003263, A033999, A067592, A239002, A294204, A054770.

CoefficientList[Series[Product[1 + x^LucasL[k], {k, 2, 15}], {x, 0, 100}], x]

G.f.: Product_{k>=2} (1 + x^Lucas(k)).

A061189 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000204(n+1), n >= 0 (Lucas numbers).

Original entry on oeis.org

1, 2, 0, -10, 15, 25, 30, 475, 450, 125, 6000, 8500, 6250, 5000, 1250, 96000, 146250, 189375, 159375, 65625, 9375, 180000, 5355000, 8881250, 5578125, 2515625, 721875, 78125, 44100000, 254700000, 341775000

Offset: 0

Wolfdieter Lang, Apr 20 2001

The row polynomials pL2(n,x) := Sum_{m=0..n} a(n,m)*x^m and pL1(n,x) := Sum_{m=0..n} A061188(n,m)*x^m appear in the k-fold convolution of the Lucas numbers L(n+1) = A000204(n+1) = A000032(n+1), n >= 0, as follows: L(k; n) := A060922(n+k,k) = (pL1(k,n)*L(n+2)+pL2(k,n)*L(n+1))/(k!*5^k).

			Triangle begins:
  {1};
  {2,0};
  {-10,15,25};
  {30,475,450,125};
  ...;
pL2(2,n) = 5*(-2+3*n+5*n^2) = 5*(1+n)*(-2+5*n).
L(2; n) := A060922(n+2,2) = A060929(n) = (1+n)*((4+5*n)*L(n+2)+(-2+5*n)*L(n+1))/(2*5).

Cf. A061188(n, m) (companion triangle), A060922(n, m) (Lucas convolution triangle).

A067980 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+2), n>=0.

Original entry on oeis.org

3, 13, 31, 69, 140, 274, 519, 963, 1757, 3165, 5642, 9972, 17499, 30521, 52955, 91461, 157336, 269702, 460863, 785295, 1334713, 2263293, 3829846, 6468264, 10905075, 18355429, 30849559, 51776133, 86785892

Offset: 0

Wolfdieter Lang, Feb 15 2002

Second diagonal of triangle A067979. Second column of triangle A067990.

a(n)= sum(L(k+1)*L(n+2-k), k=0..n) = (4*n+3)*F(n+1)+3*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).

G.f.: (3+x)*(1+2*x)/(1-x-x^2)^2.

A083564 a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204).

Original entry on oeis.org

3, 21, 72, 329, 1353, 5796, 24447, 103729, 439128, 1860621, 7880997, 33385604, 141421803, 599075421, 2537719272, 10749959329, 45537545553, 192900159396, 817138154247, 3461452823129, 14662949371128, 62113250430021

Offset: 1

Gary W. Adamson, Jun 12 2003

a(n+1)/a(n) -> (phi)^3 = ((1 + sqrt(5))/2)^3 = 4.236067...

			a(4) = Lucas(4)*Lucas(8) = 7*47 = 329.

Vincenzo Librandi, Table of n, a(n) for n = 1..160
C. Pita, On s-Fibonomials, J. Int. Seq. 14 (2011) # 11.3.7
Index entries for linear recurrences with constant coefficients, signature (3, 6, -3, -1).

Third row of array A028412.

[Lucas(n)*Lucas(2*n): n in [1..25]]; // Vincenzo Librandi, May 03 2011

Table[Fibonacci[n*4]/Fibonacci[n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

From Benoit Cloitre, Aug 30 2003: (Start)
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4);
a(n) = Fibonacci(4*n)/Fibonacci(n) = A000045(4*n)/A000045(n). (End)
a(n) = Lucas(3*n) + (-1)^n*Lucas(n).
From R. J. Mathar, Oct 27 2008: (Start)
G.f.: x*(3+12*x-9*x^2-4*x^3)/((1+x-x^2)*(1-4*x-x^2)).
a(n) = A061084(n+1) + 2*A001077(n). (End)
a(n) = (1+phi)^n + (-phi)^n + (2*phi+1)^n + (3-2*phi)^n, phi = (1+sqrt(5))/2. - Gary Detlefs, Dec 09 2012

A085292 Product of Lucas (A000204) and a Pell companion series (A001333).

Original entry on oeis.org

1, 9, 28, 119, 451, 1782, 6931, 27119, 105868, 413649, 1615681, 6311522, 24654241, 96306849, 376200748, 1469546399, 5740457491, 22423834422, 87593763331, 342165736199, 1336595027068, 5221113899769, 20395130698081, 79669083012482

Offset: 1

Gary W. Adamson, Jun 24 2003

Index entries for linear recurrences with constant coefficients, signature (2, 7, 2, -1).

Cf. A085293, A000204, A001333, A002203.

L[0] = 2; L[1] = 1; L[n] = L[n - 1] + L[n - 2]; P[0] = P[1] = 1; P[n_] := P[n] = 2P[n - 1] + P[n - 2]; Table[ L[n]P[n], {n, 1, 24}]
With[{nn=30},Rest[LinearRecurrence[{2,1},{1,1},nn]LucasL[Range[0,nn-1]]]] (* Harvey P. Dale, Apr 20 2012 *)
LinearRecurrence[{2, 7, 2, -1},{1, 9, 28, 119},24] (* Ray Chandler, Aug 03 2015 *)

a(n) = A000204(n) * A001333(n).
a(n) = 2*a(n-1)+7*a(n-2)+2*a(n-3)-a(n-4). G.f.: -x*(2*x^3-3*x^2-7*x-1) / (x^4-2*x^3-7*x^2-2*x+1). - Colin Barker, Oct 15 2013
2* A085292(n) = A085293(n).

Edited and extended by Robert G. Wilson v, Jun 24 2003

A113910 Integers of the form (Lucas(i+1) - 2*A006206(i+2))/(A006206(i+2) - A006206(i)), i > 2; Lucas = A000204.

Original entry on oeis.org

3, 7, 5, 9, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487

Offset: 1

Creighton Dement, Jan 29 2006

nonn

Let p and p+2 be twin primes. Then Lucas(p) = 1 + p*A006206(p) and Lucas(p+2) = 1 + (p+2)*A006206(p+2). It follows from Lucas(n) + Lucas(n+1) = Lucas(n+2) that p = (Lucas(p+1) - 2*A006206(p+2))/(A006206(p+2) - A006206(p))

For i = 3, 4, 5, 6, 7, 8, 9, 10, 11: ((Lucas(i+1) - 2*A006206(i+2))/(A006206(i+2) - A006206(i))) = (3, 7, 5, 19/3, 31/4, 9, 87/10, 149/14, 11, 135/11, 663/50, 1094/77, 1787/120, 2939/181, 17, 7849/434, 12799/672, 20894/1041, 34031/1622, 55469/2514, 45131/1962, 146921/6115, 238915/9554, 194252/7465, 631347/23386, 1025917/36617, 29, 2706059/90178, 4393211/141710, 3565643/111405, 11573003/350702). - Creighton Dement, Jan 31 2006

Cf. A000204, A006206, A001359.

# First 63 Terms with(combinat): with(numtheory): A006206 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/n); end; A000204 := n->fibonacci(n+1)+fibonacci(n-1); T := n -> (A000204(n+1) - 2*A006206(n+2))/(A006206(n+2)-A006206(n)); A113910 := []: for i from 3 by 1 to 2000 do if is(T(i) = floor(T(i))) then A113910 := [op(A113910), T(i)]; fi: od: A113910; # Creighton Dement, Jan 15 2009

It is conjectured that a(n+4) = A001359(n+2) for all n.

Extended and Maple definition by Creighton Dement, Jan 15 2009

A151558 Decimal expansion of the Lucas nested (A000204) radical.

Original entry on oeis.org

1, 8, 4, 0, 7, 6, 8, 3, 2, 8, 1, 4, 6, 0, 8, 2, 6, 8, 9, 8, 2, 0, 5, 7, 4, 5, 7, 7, 7, 4, 3, 5, 5, 7, 7, 8, 8, 6, 2, 1, 0, 6, 0, 3, 8, 7, 7, 7, 2, 1, 5, 4, 1, 6, 0, 5, 0, 8, 4, 9, 0, 2, 5, 0, 0, 4, 7, 8, 5, 1, 7, 4, 0, 5, 5, 6, 3, 4, 7, 0, 7, 7, 8, 5, 8, 9, 6, 0, 4, 5, 7, 0, 8, 6, 5, 1, 7, 2, 7, 8, 6, 1, 9, 5, 1

Offset: 1

Michael David Hirschhorn, May 21 2009

Analog of A105817 for the Lucas numbers (A000204).

			1.840768328146082689820574577743557788621060387772...

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse@ LucasL@ Range@ 45], 10, 111][[1]] (* Robert G. Wilson v, May 29 2009 *)

More terms from Robert G. Wilson v, May 29 2009

cp's OEIS Frontend

A166169 a(n) = Lucas(n^2) = A000204(n^2) for n >= 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A204291 G.f.: Sum_{n>=1} moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where Lucas(n) = A000204(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A225528 a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A294203 Number of partitions of n into distinct Lucas parts (A000204) greater than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A061189 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000204(n+1), n >= 0 (Lucas numbers).

Views

Author

Keywords

Comments

Examples

Crossrefs

A067980 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+2), n>=0.

Views

Author

Keywords

Comments

Formula

A083564 a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A085292 Product of Lucas (A000204) and a Pell companion series (A001333).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A204291 G.f.: Sum_{n>=1} moebius(n)x^n/(1 - Lucas(n)x^n + (-1)^nx^(2n)), where Lucas(n) = A000204(n).