A000032 -id:A000032 - cp's OEIS frontend

A039647 Related to A000032 (Lucas numbers): (n-1)!*L(n).

1, 3, 8, 42, 264, 2160, 20880, 236880, 3064320, 44634240, 722131200, 12853209600, 249559833600, 5249378534400, 118911189196800, 2886037330176000, 74715282690048000, 2055161959538688000, 59855791774851072000, 1840125433884401664000, 59547709552131440640000

Offset: 1

Wolfdieter Lang

Number of possible well-colored circuits.

C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

a(n) = A039692(n, 1) (first column of Fibonacci Jabotinsky-triangle).

Cf. A000032, A000142.

nn=19;Drop[Range[0,nn]!CoefficientList[Series[Log[1/(1-x-x^2)],{x,0,nn}],x],1] (* Geoffrey Critzer, Jul 01 2013 *)

a(n) = (n-1)!*L(n), L(n) := A000032(n); E.g.f.: -log(1-x-x^2). Also a(n)/n! = sum(binomial(n-j, j)/(n-j), j=0..floor(n/2)).

a(n) = (n-1)*(a(n-1)+(n-2)*a(n-2)), for n > 2. - Christian Krause, Oct 15 2023

A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 12, 70, 280, 905, 2568, 6666, 16220, 37580, 83780, 181074, 381488, 786715, 1593160, 3176210, 6246732, 12139859, 23344760, 44471340, 84005640, 157483176, 293201912, 542468100, 997906400, 1826073525

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 (4,-2,-8,5,8,-2,-4,-1).

Cf. A000032, A000204, A004799, A060922, A060929.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^4 )); // G. C. Greubel, Apr 08 2021

Table[((25*n^3+90*n^2+95*n+6)*LucasL[n+4] -12*(5*n^2+10*n-3)*LucasL[n+2])/150, {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

def A060930_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^4 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021

G.f.: ((1+2*x)/(1-x-x^2))^4.
a(n) = A060922(n+3, 3) (fourth column of Lucas triangle).
a(n) = (2*(25*n^3 + 60*n^2 + 35*n +24)*L(n+2) + (25*n^3 + 90*n^2 + 95*n + 6)*L(n+1))/(3!*5^2), with the Lucas numbers L(n) = A000032(n).

A075150 a(n) = L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).

Original entry on oeis.org

4, -1, 9, -16, 49, -121, 324, -841, 2209, -5776, 15129, -39601, 103684, -271441, 710649, -1860496, 4870849, -12752041, 33385284, -87403801, 228826129, -599074576, 1568397609, -4106118241, 10749957124, -28143753121, 73681302249, -192900153616, 505019158609, -1322157322201

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 05 2002

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,2,1).

Cf. A000032, A001254, A061084, A219233.

A075150:= func< n | (-1)^n*Lucas(n)^2 >; // G. C. Greubel, Jun 14 2025

CoefficientList[Series[(4 + 7*x - x^2)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 30}], x]
LinearRecurrence[{-2,2,1},{4,-1,9},50] (* Harvey P. Dale, Nov 08 2011 *)

a(n) = round((2+(1/2*(-3-sqrt(5)))^n+(1/2*(-3+sqrt(5)))^n)) \\ Colin Barker, Oct 01 2016

Vec((4+7*x-x^2)/(1+2*x-2*x^2-x^3) + O(x^30)) \\ Colin Barker, Oct 01 2016

def A075150(n): return (-1)**n*lucas_number2(n,1,-1)**2 # G. C. Greubel, Jun 14 2025

a(n) = (-1)^n*A000032(2*n) + 2.
a(n) = -2*a(n-1) + 2*a(n-2) + a(n-3) with a(0)=4, a(1)=-1, a(2)=9.
G.f.: (4 + 7*x - x^2)/(1 + 2*x - 2*x^2 - x^3).
a(n) = (-1)^n*A001254(n). - R. J. Mathar, Jan 11 2012
a(n) = 2 + (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n. - Colin Barker, Oct 01 2016
From G. C. Greubel, Jun 14 2025: (Start)
a(n) = A000032(n)*A000032(-n) = (-1)^n*A000032(n)^2.
a(n) = A219233(n) + 2 + [n=0].
E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) + 2*exp(x). (End)

A099016 a(n) = 3L(2n)/5 - (-1)^n/5, where L = A000032.

Original entry on oeis.org

1, 2, 4, 11, 28, 74, 193, 506, 1324, 3467, 9076, 23762, 62209, 162866, 426388, 1116299, 2922508, 7651226, 20031169, 52442282, 137295676, 359444747, 941038564, 2463670946, 6449974273, 16886251874, 44208781348, 115740092171

Offset: 0

Paul Barry, Sep 22 2004

Let M = an infinite triangle with (1,2,2,3,3,4,4,...) as the left border and all other columns = (0,1,2,3,4,5,...). Then lim_{n->infinity} M^n = A099016, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000032.

[3*Lucas(2*n)/5-(-1)^n/5: n in [0..35]]; // Vincenzo Librandi, Jun 09 2011

F:=Fibonacci; [F(n+1)^2+F(n)*F(n-2): n in [0..30]]; // Bruno Berselli, Feb 15 2017

with(combinat):seq(3*fibonacci(n)^2+(-1)^n, n= 0..27)

CoefficientList[Series[(1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)), {x,0,50}], x] (* G. C. Greubel, Dec 31 2017 *)

x='x+O('x^30); Vec((1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2))) \\ G. C. Greubel, Dec 31 2017

G.f.: (1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = 2*F(n)^2 + F(n)*F(n-1) + F(n-1)^2, where F = A000045.
a(n) = 3*((3/2 - sqrt(5)/2)^n + (3/2 + sqrt(5)/2)^n)/5 - (-1)^n/5.
a(n) = A099015(n)/F(n+1).
a(n) = 3*A005248(n)/5 - (-1)^n/5.
a(n) = 3*A000032(2*n)/5 - (-1)^n/5.
a(n) = A061646(n) + F(n)^2.
a(n) = 3*F(n)^2 + (-1)^n.
a(n) = F(n+1)^2 + F(n)*F(n-2). See also A192914, fourth formula. - Bruno Berselli, Feb 15 2017

A129729 Primes dividing numbers k such that k divides the k-th Lucas number A000032(k).

Original entry on oeis.org

2, 3, 107, 1283, 8747, 21401, 34667, 46187, 104003, 137387, 138563, 374929, 549547, 2204243, 2771281, 2808107, 11128427, 11223683, 13497443, 14880347, 21747529, 22753547, 23712683, 33697283, 44513387, 46970929, 57395627, 65898683

Offset: 1

Alexander Adamchuk, May 12 2007

nonn

A prime p belongs to this sequence iff for some positive integer m, m*p divides A000032(m*p) or, alternatively, m*p belongs to A016089.

The minimum multiples from A016089 of listed primes are given by A140258.

C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A016089, A140258.

Revised and extended by Max Alekseyev, May 16 2008

A153416 Decimal expansion of Sum_{n>=0} 1/A000032(2*n+1).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Dec 25 2008

			1.3966804973982612325...

Cf. A000032, A002878, A093540, A153415.

RealDigits[ NSum[ 1/LucasL[2*n + 1], {n, 0, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* Jean-François Alcover, Feb 07 2013 *)

From Amiram Eldar, Jul 05 2025: (Start)
Equals Sum_{n>=0} 1/A002878(n).
Equals A093540 - A153415. (End)

A156279 4 times the Lucas number A000032(n).

Original entry on oeis.org

8, 4, 12, 16, 28, 44, 72, 116, 188, 304, 492, 796, 1288, 2084, 3372, 5456, 8828, 14284, 23112, 37396, 60508, 97904, 158412, 256316, 414728, 671044, 1085772, 1756816, 2842588, 4599404, 7441992

Offset: 0

Paul Curtz, Feb 07 2009

This is a second kind "autosequence" whose first kind companion is A022087. - Jean-François Alcover, Aug 20 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A014217.

[4*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[4*LucasL[n], {n,0,30}] (* G. C. Greubel, Dec 21 2017 *)

a(n)=4*(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 4*A000032(n).
a(n) = a(n-1) + a(n-2).
a(n) = A014217(n+3) - A014217(n-3), with A014217(-5) = -11, A014217(-4) = 6, A014217(-3) = -4, A014217(-2) = 2, A014217(-1) = -1 extended as proposed in A153263.
G.f. 4*(-2 + x) / (-1 + x + x^2). - R. J. Mathar, Mar 11 2011
a(n) = Lucas(n+3) - Lucas(n-3), where Lucas(i) for i = 0..2 gives -4, 3, -1. - Bruno Berselli, Jul 27 2017

A191916 Ordered sums f+2g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 29, 31, 32, 33, 35, 37, 39, 40, 43, 47, 49, 51, 53, 54, 55, 59, 61, 62, 65, 69, 76, 78, 82, 83, 84, 87, 90, 95, 97, 98, 101, 105, 112, 123, 125, 129, 131, 134, 137, 141, 145, 153, 155

Offset: 1

Clark Kimberling, Jun 19 2011

nonn

Cf. A191917, A191918, A191919, A191838.

c = 1; d = 2; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u ]]    (* A191916 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1 ]]  (* A191917: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2 ]]  (* A191918: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]         (* A191919*)

A191920 Ordered sums f+3*g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

4, 6, 7, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 32, 34, 36, 37, 38, 39, 40, 41, 44, 50, 51, 55, 56, 57, 58, 59, 61, 62, 65, 68, 72, 79, 80, 83, 85, 88, 90, 91, 94, 97, 98, 101, 105, 109, 116, 126, 130, 132, 134, 135, 142, 144, 145

Offset: 1

Clark Kimberling, Jun 19 2011

nonn

Cf. A000032, A191921, A191922, A191923, A191842.

c = 1; d = 3; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A191920 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A191921: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A191922: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A191923 *)

A191921 Ordered sequence of nonnegative differences f-3*g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 14, 15, 17, 20, 22, 26, 35, 36, 38, 43, 44, 55, 58, 64, 67, 69, 73, 90, 94, 102, 111, 112, 114, 120, 145, 152, 166, 178, 181, 187, 190, 196, 235, 246, 268, 289, 293, 301, 310, 313, 319, 380, 398, 434, 467, 474, 488, 500, 509, 512, 518

Offset: 1

Clark Kimberling, Jun 19 2011

nonn

Cf. A191920, A191922, A191923.

Mathematica
```
(See A191920.)
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cp's OEIS Frontend

A039647 Related to A000032 (Lucas numbers): (n-1)!*L(n).

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A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

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A075150 a(n) = L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).

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A099016 a(n) = 3*L(2*n)/5 - (-1)^n/5, where L = A000032.

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A129729 Primes dividing numbers k such that k divides the k-th Lucas number A000032(k).

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A153416 Decimal expansion of Sum_{n>=0} 1/A000032(2*n+1).

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A156279 4 times the Lucas number A000032(n).

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A191916 Ordered sums f+2g, where f and g are Lucas numbers (A000032 beginning at 1).

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A099016 a(n) = 3L(2n)/5 - (-1)^n/5, where L = A000032.