A000032 -id:A000032 - cp's OEIS frontend

A215602 a(n) = L(n)*L(n+1), where L = A000032 (Lucas numbers).

2, 3, 12, 28, 77, 198, 522, 1363, 3572, 9348, 24477, 64078, 167762, 439203, 1149852, 3010348, 7881197, 20633238, 54018522, 141422323, 370248452, 969323028, 2537720637, 6643838878, 17393796002, 45537549123, 119218851372, 312119004988, 817138163597, 2139295485798, 5600748293802, 14662949395603, 38388099893012, 100501350283428

Offset: 0

N. J. A. Sloane, Aug 17 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Ömer Eğecioğlu, Elif Saygı, and Zülfükar Saygı, The Mostar and Wiener index of alternate Lucas cubes, Transactions on Combinatorics (2023) Vol. 12, No. 1, 37-46.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000032, A215580. A075269 is a signed version.

Table[LucasL[n]*LucasL[n + 1], {n, 0, 33}] (* Amiram Eldar, Oct 06 2020 *)

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

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

G.f.: ( 2-x+2*x^2 ) / ( (1+x)*(x^2-3*x+1) ). - R. J. Mathar, Aug 21 2012
a(n) = A002878(n)+(-1)^n. - R. J. Mathar, Aug 21 2012
a(n) = F(n-1)*F(n) + F(n-1)*F(n+2) + F(n+1)*F(n) + F(n+1)*F(n+2), where F=A000045, F(-1)=1. - Bruno Berselli, Nov 03 2015
a(n) = F(2*n) + F(2*n+2) + (-1)^n with F(k)=A000045(k). - J. M. Bergot, Apr 15 2016
a(n) = ((-1)^n+(2^(-1-n)*((3-sqrt(5))^n*(-5+sqrt(5))+(3+sqrt(5))^n*(5+sqrt(5)))) / sqrt(5)). - Colin Barker, Oct 01 2016
Sum_{n>=0} (-1)^n/a(n) = sqrt(5)/10. - Amiram Eldar, Oct 06 2020

A067593 Number of partitions of n into Lucas parts (A000032).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 12, 16, 20, 26, 33, 41, 50, 62, 75, 90, 107, 129, 151, 178, 208, 244, 281, 326, 375, 431, 491, 561, 638, 723, 816, 922, 1037, 1163, 1302, 1458, 1624, 1808, 2009, 2231, 2467, 2729, 3012, 3321, 3651, 4014, 4406, 4828, 5282, 5777, 6308, 6877, 7491, 8155, 8862, 9622, 10438, 11316, 12247, 13249

Offset: 0

Naohiro Nomoto, Jan 31 2002

			a(5)=6 because we have 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1 and 1+1+1+1+1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

N=66; q='q+O('q^N);
L(n) = fibonacci(n+1) + fibonacci(n-1);
gf = 1; k=0; while( L(k) <= N, gf*=(1-q^L(k)); k+=1 ); gf = 1/gf;
Vec( gf ) /* Joerg Arndt, Mar 26 2014 */

G.f.: 1/((1-x^2)*prod(i>=1, 1-x^(fibonacci(i-1)+fibonacci(i+1)) ) ). - Emeric Deutsch, Mar 23 2005

G.f.: 1 / prod(n>=0, 1 - q^A000032(n) ). [Joerg Arndt, Mar 26 2014]

A121171 Largest prime divisor of Lucas(5*n), where Lucas(k) = A000032(k).

Original entry on oeis.org

11, 41, 31, 2161, 151, 2521, 911, 3041, 541, 570601, 39161, 20641, 24571, 12317523121, 18451, 23725145626561, 12760031, 10783342081, 87382901, 5738108801, 767131, 59996854928656801, 686551, 23735900452321, 28143378001, 42426476041450801, 119611

Offset: 1

Alexander Adamchuk, Aug 14 2006

nonn

Final digit of a(n) is 1. Mod[a(n),10] = 1. Final digit of many prime divisors of Lucas(5*n) is 1.

Daniel Suteu, Table of n, a(n) for n = 1..309

Cf. A000032, A000045, A079451.

Table[Max[Flatten[FactorInteger[Fibonacci[5n-1]+Fibonacci[5n+1]]]],{n,1,40}]

lucas(n) = fibonacci(n+1)+fibonacci(n-1); \\ A000032
a(n) = vecmax(factor(lucas(5*n))[,1]); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A000032(5*n)) = A079451(5*n). - Daniel Suteu, May 26 2022

A166473 a(n) = 2^L(n+1) * 3^L(n)/12, where L(n) is the n-th Lucas number (A000032(n)).

Original entry on oeis.org

2, 36, 864, 373248, 3869835264, 17332899271409664, 804905577934332296851095552, 167416167663978753511691999938432197602574336

Offset: 1

Matthew Vandermast, Nov 05 2009

nonn

For m>1, A166469(A002110(m)*a(n)) = L(m+n).

A166469(a(n)) = L(n+2) - 2 = A014739(n).

G. C. Greubel, Table of n, a(n) for n = 1..14

Subsequence of A003586, A025487.

List([1..10], n-> 2^(Lucas(1,-1,n+1)[2]-2)*3^(Lucas(1,-1,n)[2]-1)); # G. C. Greubel, Jul 22 2019

[2^(Lucas(n+1)-2)*3^(Lucas(n)-1): n in [1..10]]; // G. C. Greubel, Jul 22 2019

Table[(2^LucasL[n+1] 3^LucasL[n])/12,{n,10}] (* Harvey P. Dale, Aug 17 2011 *)

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
vector(10, n,  2^(lucas(n+1)-2)*3^(lucas(n)-1) ) \\ G. C. Greubel, Jul 22 2019

[2^(lucas_number2(n+1,1,-1)-2)*3^(lucas_number2(n,1,-1)-1) for n in (1..10)] # G. C. Greubel, Jul 22 2019

a(n) = A166471(n)/12.

For n>1, a(n) = 12*a(n-1) * a(n-2).

A191929 Ordered sums f+4g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

5, 7, 8, 11, 13, 15, 16, 17, 19, 20, 22, 23, 27, 29, 30, 31, 32, 33, 34, 35, 39, 41, 45, 46, 47, 48, 51, 55, 57, 59, 62, 63, 73, 75, 76, 79, 80, 83, 88, 90, 91, 92, 101, 104, 117, 119, 120, 123, 127, 134, 135, 139, 145, 148, 151, 163, 167, 189, 191, 192, 195

Offset: 1

Clark Kimberling, Jun 19 2011

nonn

Cf. A000032, A191932, A191931, A191932, A191850.

c = 1; d = 4; 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]]    (* A191929 *)
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]]  (* A191930: 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]]  (* A191931: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A191932 *)

A203318 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(nk)x^(n*k)/k ) ) where Lucas(n) = A000032(n).

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 36, 64, 135, 250, 504, 917, 1864, 3372, 6593, 12176, 23473, 42732, 82142, 149282, 283104, 516780, 967894, 1757865, 3291964, 5959633, 11039163, 20022457, 36908442, 66637739, 122512809, 220717328, 403499293, 726866565, 1322670966, 2376541137

Offset: 0

Paul D. Hanna, Dec 31 2011

nonn

Note: 1/(1-x-x^2) = exp(Sum_{n>=1} Lucas(n)*x^n/n) is the g.f. of the Fibonacci numbers.

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} F_n(x^n) * x^n/n )
where F_n(x) = exp( Sum_{k>=1} Lucas(n*k)*x^k/k ), which begin:
F_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 +...;
F_2(x) = 1 + 3*x + 8*x^2 + 21*x^3 + 55*x^4 + 144*x^5 + 377*x^6 +...;
F_3(x) = 1 + 4*x + 17*x^2 + 72*x^3 + 305*x^4 + 1292*x^5 + 5473*x^6 +...;
F_4(x) = 1 + 7*x + 48*x^2 + 329*x^3 + 2255*x^4 + 15456*x^5 +...;
F_5(x) = 1 + 11*x + 122*x^2 + 1353*x^3 + 15005*x^4 + 166408*x^5 +...;
F_6(x) = 1 + 18*x + 323*x^2 + 5796*x^3 + 104005*x^4 + 1866294*x^5 +...;
...
Also, F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where u = n-th root of unity:
F_1(x) = F(x) = 1/(1-x-x^2) = g.f. of the Fibonacci numbers;
F_2(x^2) = F(x)*F(-x) = 1/(1-3*x^2+x^4);
F_3(x^3) = F(x)*F(w*x)*F(w^2*x) = 1/(1-4*x^3-x^6) where w = exp(2*Pi*I/3);
F_4(x^4) = F(x)*F(I*x)*F(-x)*F(-I*x) = 1/(1-7*x^4+x^8);
F_5(x^5) = 1/(1-11*x^5-x^10);
In general,
F_n(x^n) = 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
...
The logarithmic derivative of this sequence begins:
A203319 = [1,3,7,19,26,81,92,267,358,848,980,3061,3030,7976,...].

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A203319, A203320, A000032 (Lucas); A203413 (Pell variant).

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

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=vector(n+1, i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)}

{a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}

G.f.: exp( Sum_{n>=1} A203319(n)*x^n/n ) where A203319(n) = n*fibonacci(n)*Sum_{d|n} 1/(d*fibonacci(d)).
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) ) where Lucas(n) = A000032(n).
G.f.: exp( Sum_{n>=1} F_n(x^n) * x^n/n ) such that F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where F(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A263575 Stirling transform of Lucas numbers (A000032).

Original entry on oeis.org

2, 1, 4, 14, 53, 227, 1092, 5791, 33350, 206511, 1365563, 9590847, 71216713, 556861216, 4569168866, 39222394456, 351304769679, 3275433717440, 31723522878974, 318571978752719, 3311400814816987, 35573458376435132, 394404160256111139, 4507130777468928696

Offset: 0

Vladimir Reshetnikov, Oct 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..563
Eric Weisstein's MathWorld, Lucas Number.
Eric Weisstein's MathWorld, Stirling Transform.
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A000032, A213593, A005248, A061084, A263576.

Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]

a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).
Let phi = (1+sqrt(5))/2.
a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.
2*B_n(phi) = a(n) + A263576*sqrt(5).
E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).
Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).
G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

A280694 Largest Lucas number (A000032) dividing n.

Original entry on oeis.org

1, 2, 3, 4, 1, 3, 7, 4, 3, 2, 11, 4, 1, 7, 3, 4, 1, 18, 1, 4, 7, 11, 1, 4, 1, 2, 3, 7, 29, 3, 1, 4, 11, 2, 7, 18, 1, 2, 3, 4, 1, 7, 1, 11, 3, 2, 47, 4, 7, 2, 3, 4, 1, 18, 11, 7, 3, 29, 1, 4, 1, 2, 7, 4, 1, 11, 1, 4, 3, 7, 1, 18, 1, 2, 3, 76, 11, 3, 1, 4, 3, 2, 1, 7, 1, 2, 29, 11, 1, 18, 7, 4, 3, 47, 1, 4, 1, 7, 11, 4, 1, 3, 1, 4, 7, 2, 1, 18, 1, 11, 3, 7, 1

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15127

Cf. A000032, A054494, A280695, A280699.

Cf. A057854 (gives the positions n > 1 where this sequence and A280696 obtain equal values).

;; A stand-alone program:
(define (A280694 n) (let loop ((l1 1) (l2 3) (lpd 1)) (cond ((> l1 n) (if (and (= 1 lpd) (even? n)) 2 lpd)) ((zero? (modulo n l1)) (loop l2 (+ l1 l2) l1)) (else (loop l2 (+ l1 l2) lpd)))))

a(n) = n / A280695(n).
Other identities. For all n >= 1:
a(A000032(n)) = A000032(n).
a(A057854(n)) = A280696(A057854(n)).
a(A000045(n)) = A280699(n).

A355021 a(n) = (-1)^n * A000032(n) - 1.

Original entry on oeis.org

1, -2, 2, -5, 6, -12, 17, -30, 46, -77, 122, -200, 321, -522, 842, -1365, 2206, -3572, 5777, -9350, 15126, -24477, 39602, -64080, 103681, -167762, 271442, -439205, 710646, -1149852, 1860497, -3010350, 4870846, -7881197, 12752042, -20633240, 33385281

Offset: 0

Clark Kimberling, Jun 21 2022

There are the partial sums of L(1) - L(2) + L(3) - L(4) + L(5) - ... .
Closely related (Fibonacci, A000045) partial sums of F(1) - F(2) + F(3) - F(4) + F(5) - ... are given by A355020.
Apart from signs, same as A098600 and A181716.

			a(0) = 1;
a(1) = 1 - 3 = -2;
a(2) = 1 - 3 + 4 = 2;
a(3) = 1 - 3 + 4 - 7 = -5.

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

Cf. A000032, A000045, A098600, A181716, A355018, A355019, A355020.

[Lucas(-n) -1: n in [0..50]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1 = Table[(-1)^n f[n] + 1, {n, 0, 40}]   (* A355020 *)
g1 = Table[(-1)^n g[n] - 1, {n, 0, 40}]   (* this sequence *)
LucasL[-Range[0, 50]] - 1 (* G. C. Greubel, Mar 17 2024 *)
LinearRecurrence[{0,2,-1},{1,-2,2},40] (* Harvey P. Dale, Sep 06 2024 *)

[lucas_number2(-n,1,-1) -1 for n in range(51)] # G. C. Greubel, Mar 17 2024

a(n) = 2*a(n-2) - a(n-3) for n >= 3. [Corrected by Georg Fischer, Sep 30 2022]

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

A380821 Length of the shorts leg in the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

5, 3, 7, 9, 15, 23, 37, 59, 95, 153, 247, 399, 645, 1043, 1687, 2729, 4415, 7143, 11557, 18699, 30255, 48953, 79207, 128159, 207365, 335523, 542887, 878409, 1421295, 2299703, 3720997, 6020699, 9741695, 15762393, 25504087, 41266479, 66770565, 108037043

Offset: 0

Miguel-Ángel Pérez García-Ortega, Feb 04 2025

			 n=0:      5,    12,    13;
 n=1:      3,     4,     5;
 n=2:      7,    24,    25;
 n=3:      9,    40,    41.
This sequence is the first column.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Cf. A380823 (semiperimeter), A380824 (area), A000032 (inradius), A386201 (long legs).

a=Table[LucasL[n],{n,0,15}];Apply[Join,Map[{2#+1,2#^2+2#,2#^2+2#+1}&,a]]

a(n) = 2*A000032(n) + 1.

cp's OEIS Frontend

A215602 a(n) = L(n)*L(n+1), where L = A000032 (Lucas numbers).

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A067593 Number of partitions of n into Lucas parts (A000032).

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A121171 Largest prime divisor of Lucas(5*n), where Lucas(k) = A000032(k).

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A166473 a(n) = 2^L(n+1) * 3^L(n)/12, where L(n) is the n-th Lucas number (A000032(n)).

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A191929 Ordered sums f+4g, where f and g are Lucas numbers (A000032 beginning at 1).

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A203318 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n).

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A263575 Stirling transform of Lucas numbers (A000032).

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A280694 Largest Lucas number (A000032) dividing n.

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A203318 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(nk)x^(n*k)/k ) ) where Lucas(n) = A000032(n).