A061084 -id:A061084 - cp's OEIS frontend

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

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)

A075091 Sum of Lucas numbers and reflected Lucas numbers (comment to A061084).

Original entry on oeis.org

4, 0, 6, 0, 14, 0, 36, 0, 94, 0, 246, 0, 644, 0, 1686, 0, 4414, 0, 11556, 0, 30254, 0, 79206, 0, 207364, 0, 542886, 0, 1421294, 0, 3720996, 0, 9741694, 0, 25504086, 0, 66770564, 0, 174807606, 0, 457652254, 0, 1198149156, 0, 3136795214, 0, 8212236486, 0, 21499914244, 0

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 31 2002

a(n) = ((-1)^n + 1)*L(n), where L(n) denotes the n-th Lucas number.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000032, A061084.

CoefficientList[Series[(4 - 6*x^2)/(1 - 3*x^2 + x^4), {x, 0, 50}], x]
LinearRecurrence[{0,3,0,-1},{4,0,6,0},50] (* or *) Riffle[ LinearRecurrence[ {3,-1},{4,6},30],0] (* Harvey P. Dale, Aug 17 2018 *)

a(n) = 3*a(n-2) - a(n-4), a(0)=4, a(1)=0, a(2)=6, a(3)=0.
G.f.: (4 - 6x^2)/(1 - 3*x^2 + x^4).
E.g.f.: 4*cosh(x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Aug 30 2025

a(46)-a(49) from Stefano Spezia, Aug 30 2025

A075151 a(n)=L(n)^2*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (comment to A061084).

Original entry on oeis.org

8, -1, 27, -64, 343, -1331, 5832, -24389, 103823, -438976, 1860867, -7880599, 33386248, -141420761, 599077107, -2537716544, 10749963743, -45537538411, 192900170952, -817138135549, 3461452853383, -14662949322176, 62113250509227, -263115950765039, 1114577054530568

Offset: 0

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,6,3,-1).

Cf. A000032, A061084, A001254, A075155.

[((-1)^n*Lucas(n))^3: n in [0..30]]; // Vincenzo Librandi, Apr 22 2018

CoefficientList[Series[(8+23*x-24*x^2-x^3)/(1+3*x-6*x^2-3*x^3+x^4), {x, 0, 25}], x]
LinearRecurrence[{-3,6,3,-1},{8,-1,27,-64},30] (* Harvey P. Dale, Apr 06 2013 *)
Table[LucasL[-n]^3, {n, 0, 25}] (* Vincenzo Librandi, Apr 22 2018 *)

a(n) = 3*L(n)+(-1)^n*L(3n).
a(n) = -3a(n-1)+6a(n-2)+3a(n-3)-a(n-4), n>3.
G.f.: ( 8+23*x-24*x^2-x^3 ) / ( (x^2+x-1)*(x^2-4*x-1) ).
a(n) is asymptotic to (-phi)^(3n) where phi is the golden ratio (1+sqrt(5))/2. - Benoit Cloitre, Sep 07 2002
a(n) = ((-1)^n*L(n))^3 = L(-n)^3. - Ehren Metcalfe, Apr 21 2018

A124844 Triangle T(n,k)=binomial(n,k)*A061084(k), 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 4, -1, 1, 6, -3, 3, 1, 8, -6, 12, -4, 1, 10, -10, 30, -20, 7, 1, 12, -15, 60, -60, 42, -11, 1, 14, -21, 105, -140, 147, -77, 18, 1, 16, -28, 168, -280, 392, -308, 144, -29, 1, 18, -36, 252, -504, 882, -924, 648, -261, 47, 1, 20, -45, 360, -840, 1764, -2310, 2160, -1305, 470, -76

Offset: 0

Gary W. Adamson, Nov 10 2006

			First few rows of the triangle are:
1;
1, 2;
1, 4, -1;
1, 6, -3, 3;
1, 8, -6, 12, -4;
1, 10, -10, 30, -20, 7;
1, 12, -15, 60, -60, 42, -11;
1, 14, -21, 105, -140, 147, -77, 18;
...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A061084, A000032, A000204 (row sums).

Cf. A007318.

import Data.List (inits)
a124844 n k = a124844_tabl !! n !! k
a124844_row n = a124844_tabl !! n
a124844_tabl = zipWith (zipWith (*))
                       a007318_tabl $ tail $ inits a061084_list
-- Reinhard Zumkeller, Sep 15 2015

We let A061084 = the diagonal of an infinite matrix, M. Perform P*M and extract the zeros, where P = Pascal's triangle as an infinite lower triangular matrix.

A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

Original entry on oeis.org

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324

Offset: 1

N. J. A. Sloane

See A000032 for the version beginning 2, 1, 3, 4, 7, ...
Also called Schoute's accessory series (see Jean, 1984). - N. J. A. Sloane, Jun 08 2011
L(n) is the number of matchings in a cycle on n vertices: L(4)=7 because the matchings in a square with edges a, b, c, d (labeled consecutively) are the empty set, a, b, c, d, ac and bd. - Emeric Deutsch, Jun 18 2001
This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1..m - 1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n - 1 - (m - 1)*i, i - 1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m = 2: A000204, m = 3: A001609, m = 4: A014097, m = 5: A058368, m = 6: A058367, m = 7: A058366, m = 8: A058365, m = 9: A058364.
L(n) is the number of points of period n in the golden mean shift. The number of orbits of length n in the golden mean shift is given by the n-th term of the sequence A006206. - Thomas Ward, Mar 13 2001
Row sums of A029635 are 1, 1, 3, 4, 7, ... - Paul Barry, Jan 30 2005
a(n) counts circular n-bit strings with no repeated 1's. E.g., for a(5): 00000 00001 00010 00100 00101 01000 01001 01010 10000 10010 10100. Note #{0...} = fib(n+1), #{1...} = fib(n-1), #{000..., 001..., 100...} = a(n-1), #{010..., 101...} = a(n-2). - Len Smiley, Oct 14 2001
Row sums of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
If p is prime then L(p) == 1 (mod p). L(2^k) == -1 (mod 2^(k+1)) for k = 0,1,2,... - Thomas Ordowski, Sep 25 2013
Satisfies Benford's law [Brown-Duncan, 1970; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 08 2017

			G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 18*x^6 + 29*x^7 + 47*x^8 + ...

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
Leonhard Euler, Introductio in analysin infinitorum (1748), sections 216 and 229.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5. - N. J. A. Sloane, Jun 08 2011
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

Indranil Ghosh, Table of n, a(n) for n = 1..4780 (terms 1..500 computed by N. J. A. Sloane)
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Arno Berger and Theodore P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
J. Brown and R. L. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quarterly 8 (1970) 482-486.
Enrico Di Cera and Yong Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
Sergio Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
Scott Garrabrant and Igor Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Sarah H. Holliday and Takao Komatsu, On the Sum of Reciprocal Generalized Fibonacci Numbers, Integers. Volume 11, Issue 4, Pages 441-455.
R. Jovanovic, First 70 Lucas numbers
Blair Kelly, Factorizations of Lucas numbers
Tanya Khovanova, Recursive Sequences
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
Ron Knott, The Lucas Numbers in Pascal's Triangle.
Kantaphon Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1.
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Mathilde Noual, Dynamics in parallel of double Boolean automata circuits, arXiv:1011.3930 [cs.DM], 2010. - _N. J. A. Sloane_, Jul 07 2012
Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444. - _N. J. A. Sloane_, Jul 07 2012
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
Mark A. Shattuck and Carl G. Wagner, Periodicity and Parity Theorems for a Statistic on r-Mino Arrangements, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.6.
N. J. A. Sloane, Illustration of initial terms: the Lucas tree
Zdzisław Skupień, Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets, Discussiones Mathematicae Graph Theory. Volume 33, Issue 1, Pages 217-229, ISSN (Print) 2083-5892, DOI: 10.7151/dmgt.1658, April 2013.
Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Lucas n-Step Number
Richard J. Yanco, Letter and Email to N. J. A. Sloane, 1994
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).
Index entries for sequences related to Benford's law

Cf. A000032, A000045, A061084, A027960, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A006206, A101033, A101032, A100492, A099731, A094216, A094638, A000108, A090946 (complement).

a000204 n = a000204_list !! n
a000204_list = 1 : 3 : zipWith (+) a000204_list (tail a000204_list)
-- Reinhard Zumkeller, Dec 18 2011

[Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 17 2017

A000204 := proc(n) option remember; if n <=2 then 2*n-1; else procname(n-1)+procname(n-2); fi; end;
with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1);
# alternative Maple program:
L:= n-> (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(L(n), n=1..50);  # Alois P. Heinz, Jul 25 2008
# Alternative:
a := n -> `if`(n=1, 1, `if`(n=2, 3, hypergeom([(1-n)/2, -n/2], [1-n], -4))):
seq(simplify(a(n)), n=1..39); # Peter Luschny, Sep 03 2019

c = (1 + Sqrt[5])/2; Table[Expand[c^n + (1 - c)^n], {n, 30}] (* Artur Jasinski, Oct 05 2008 *)
Table[LucasL[n, 1], {n, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
LinearRecurrence[{1, 1},{1, 3}, 50] (* Sture Sjöstedt, Nov 28 2011 *)
lukeNum[n_] := If[n < 1, 0, LucasL[n]]; (* Michael Somos, May 18 2015 *)
lukeNum[n_] := SeriesCoefficient[x D[Log[1 / (1 - x - x^2)], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)

A000204(n)=fibonacci(n+1)+fibonacci(n-1) \\ Michael B. Porter, Nov 05 2009

from functools import cache
@cache
def a(n): return [1, 3][n-1] if n < 3 else a(n-1) + a(n-2)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Nov 13 2022

[(i:=-1)+(j:=2)] + [(j:=i+j)+(i:=j-i) for  in range(100)] # _Jwalin Bhatt, Apr 02 2025

def A000204():
    x, y = 1, 2
    while true:
       yield x
       x, y = x + y, x
a = A000204(); print([next(a) for i in range(39)])  # Peter Luschny, Dec 17 2015

def lucas(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 2, 1)
}
(1 to 50).map(lucas()) // _Alonso del Arte, Oct 20 2019

Expansion of x(1 + 2x)/(1 - x - x^2). - Simon Plouffe, dissertation 1992; multiplied by x. - R. J. Mathar, Nov 14 2007
a(n) = A000045(2n)/A000045(n). - Benoit Cloitre, Jan 05 2003
For n > 1, L(n) = F(n + 2) - F(n - 2), where F(n) is the n-th Fibonacci number (A000045). - Gerald McGarvey, Jul 10 2004
a(n+1) = 4*A054886(n+3) - A022388(n) - 2*A022120(n+1) (a conjecture; note that the above sequences have different offsets). - Creighton Dement, Nov 27 2004
a(n) = Sum_{k=0..floor((n+1)/2)} (n+1)*binomial(n - k + 1, k)/(n - k + 1). - Paul Barry, Jan 30 2005
L(n) = A000045(n+3) - 2*A000045(n). - Creighton Dement, Oct 07 2005
L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson, Sep 29 2007
a(n) = 2*Fibonacci(n-1) + Fibonacci(n), n >= 1. - Zerinvary Lajos, Oct 05 2007
L(n) is the term (1, 1) in the 1 X 2 matrix [2, -1].[1, 1; 1, 0]^n. - Alois P. Heinz, Jul 25 2008
a(n) = phi^n + (1 - phi)^n = phi^n + (-phi)^(-n) = ((1 + sqrt(5))^n + (1 - sqrt(5))^n)/(2^n) where phi is the golden ratio (A001622). - Artur Jasinski, Oct 05 2008
a(n) = A014217(n+1) - A014217(n-1). See A153263. - Paul Curtz, Dec 22 2008
a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + ((1 + sqrt(5))^(n - 1) - (1 - sqrt(5))^(n - 1))/(2^(n - 2)*sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009, Jan 14 2009
From Hieronymus Fischer, Oct 20 2010 (Start)
Continued fraction for n odd: [L(n); L(n), L(n), ...] = L(n) + fract(Fib(n) * phi).
Continued fraction for n even: [L(n); -L(n), L(n), -L(n), L(n), ...] = L(n) - 1 + fract(Fib(n)*phi). Also: [L(n) - 2; 1, L(n) - 2, 1, L(n) - 2, ...] = L(n) - 2 + fract(Fib(n)*phi). (End)
INVERT transform of (1, 2, -1, -2, 1, 2, ...). - Gary W. Adamson, Mar 07 2012
L(2n - 1) = floor(phi^(2n - 1)); L(2n) = ceiling(phi^(2n)). - Thomas Ordowski, Jun 15 2012
a(n) = hypergeom([(1 - n)/2, -n/2], [1 - n], -4) for n >= 3. - Peter Luschny, Sep 03 2019
E.g.f.: 2*(exp(x/2)*cosh(sqrt(5)*x/2) - 1). - Stefano Spezia, Jul 26 2022

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

A203976 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=4.

Original entry on oeis.org

0, 1, 5, 4, 15, 11, 40, 29, 105, 76, 275, 199, 720, 521, 1885, 1364, 4935, 3571, 12920, 9349, 33825, 24476, 88555, 64079, 231840, 167761, 606965, 439204, 1589055, 1149851, 4160200, 3010349, 10891545, 7881196, 28514435, 20633239, 74651760, 54018521, 195440845

Offset: 0

Michael Somos, Jan 08 2012

a(n+1) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = Lucas(k) for k = 1, ..., n+1.
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
a(n) = row sums of triangle A226377(n), based on differences among Lucas Numbers.  - Richard R. Forberg, Aug 01 2013
A strong divisibility sequence, i.e., gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -5, 1, -5, ...] = 1/(1 - 1/(5 - 1/(1 - 1/(5 - ...)))) = 1/2*(5 - sqrt(5)) begins [0/1, 1/1, 5/4, 4/3, 15/11, 11/8, 40/29,...]. The present sequence is the sequence of numerators; the sequence of denominators [1, 1, 4, 3, 11, 8, 29,...] is A005013. - Peter Bala, May 19 2014
It appears that the first homology group of the branched n-th cyclic covering of the group of figure-eight knot is the direct sum of cyclic groups of orders a(n) and A005013(n), so the order of that group is the product of these numbers, i. e. A004146(n); see the table on p. 156 of the paper by Fox. - Andrey Zabolotskiy, Mar 16 2023

			a(3) = 4 since p(x) = (-x^2 + 7*x - 4) / 2 interpolates p(1) = 1, p(2) = 3, p(3) = 4, and p(4) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. H. Fox, A quick trip through knot theory, pages 120-167 in: Topology of 3-manifolds and related topics (Proceedings of The University of Georgia Institute, 1961), Prentice-Hall, 1962.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000032, A000045, A201157 (bisection), A002878 (bisection). A005013.

a203976 n = a203976_list !! n
a203976_list = 0 : 1 : 5 : 4 : zipWith (-)
   (map (* 3) $ drop 2 a203976_list) a203976_list
-- Reinhard Zumkeller, Jan 10 2012

I:=[0,1,5,4]; [n le 4 select I[n] else 3*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 29 2016

LinearRecurrence[{0,3,0,-1},{0,1,5,4},40] (* Harvey P. Dale, Apr 06 2013 *)

{a(n) = if( n%2, fibonacci(n+1) + fibonacci(n-1), 5 * fibonacci(n))}

{a(n) = if( n<0, -a(-n), polcoeff( x * (1 + 5*x + x^2) / (1 - 3*x^2 + x^4) + x * O(x^n), n))}

{a(n) = if( n<0, -a(-n), subst( polinterpolate( vector( n, k, fibonacci(k-1) + fibonacci(k+1) )), x, n + 1))}

a(1) = 1, a(2) = 5, a(3) = 4, a(n) * a(n-3) = a(n-1) * a(n-2) - 5. a(-n) = -a(n).
G.f.: x * (1 + 5*x + x^2) /  ( (x^2+x-1)*(x^2-x-1) ).
a(2*n) = 5 * A000045(2*n) (Fibonacci). a(2*n+1) = A000032(2*n+1) (Lucas).
a(A004277(n)) = A054888(n+1). - Reinhard Zumkeller, Jan 11 2012
a(n) = A000032(n+1) - A061084(n). - R. J. Mathar, Jun 23 2013
a(2n) = a(2n-1) + a(2n+1), for n>0. - Richard R. Forberg, Aug 01 2013
a(n) = (2^(-1-n)*((-5-sqrt(5)+(-1)^n*(-5+sqrt(5)))*((-1+sqrt(5))^n-(1+sqrt(5))^n)))/sqrt(5). - Colin Barker, Mar 28 2016
E.g.f.: exp(-phi*x)*(exp(x) - 1)*(phi*exp(sqrt(5)*x) - 1/phi), where phi = (1 + sqrt(5))/2. - G. C. Greubel, Mar 28 2016

A075155 Cubes of Lucas numbers.

Original entry on oeis.org

8, 1, 27, 64, 343, 1331, 5832, 24389, 103823, 438976, 1860867, 7880599, 33386248, 141420761, 599077107, 2537716544, 10749963743, 45537538411, 192900170952, 817138135549, 3461452853383, 14662949322176, 62113250509227, 263115950765039, 1114577054530568

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 06 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000032, A061084, A075150, A075151.

Third row of array A103324.

[ Lucas(n)^3 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011

CoefficientList[Series[(8 - 23*x - 24*x^2 + x^3)/((x^2 + 4*x - 1)*(x^2 - x - 1)), {x,0,50}], x] (* or *) Table[LucasL[n]^3, {n,0,30}] (* or *) LinearRecurrence[{3,6,-3,-1}, {8, 1, 27, 64}, 30] (* G. C. Greubel, Dec 21 2017 *)

a(n)=(fibonacci(n-1)+fibonacci(n+1))^3 \\ Charles R Greathouse IV, Feb 09 2016

from sympy import lucas
def a(n):  return lucas(n)**3
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 01 2021

a(n) = 3*(-1)^n*L(n) + L(3*n).
a(n) = (-1)^n*A075151(n).
a(n) = A000032(n)^3 = A000032(n) * A001254(n).
a(n) = L(n)*C(n)^2, L(n) = Lucas numbers (A000032), C(n) = reflected Lucas numbers (comment to A061084).
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4), n>=4.
G.f.: ( 8-23*x-24*x^2+x^3 )/( (x^2+4*x-1)*(x^2-x-1) ).
a(n) = 2*A001077(n) + 3*A061084(n+1). - R. J. Mathar, Nov 17 2011
a(n) = L(3*n) + (F(n+4) - F(n-4))*(-1)^n, n>3 and F(n)=A000045(n). - J. M. Bergot, Feb 09 2016
a(n) + Sum_{i=0..n+1} a(i) = 19/2 + (5/2)*L(3*n+2). - Greg Dresden, Feb 24 2025

Simpler definition from Ralf Stephan, Nov 01 2004

A074062 Reflected (see A074058) pentanacci numbers A074048.

Original entry on oeis.org

5, -1, -1, -1, -1, 9, -7, -1, -1, -1, 19, -23, 5, -1, -1, 39, -65, 33, -7, -1, 79, -169, 131, -47, 5, 159, -417, 431, -225, 57, 313, -993, 1279, -881, 339, 569, -2299, 3551, -3041, 1559, 799, -5167, 9401, -9633, 6159, 39, -11133, 23969, -28667, 21951, -6081, -22305

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 17 2002

a(n) is also the trace of A^(-n), where A is the matrix ( (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1), (1,0,0,0,0) ).

a(n) is also the sum of determinants of 4th-order principal minors of A^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,1).

Cf. A061084, A074058, A074048.

I:=[5,-1,-1,-1,-1]; [n le 5 select I[n] else (-1)*(Self(n-1) +Self(n-2) +Self(n-3) +Self(n-4)) + Self(n-5): n in [1..61]]; // G. C. Greubel, Jul 05 2021

CoefficientList[Series[(5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5), {x, 0, 60}], x]

Vec((5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) + O(x^60)) \\ Michel Marcus, Sep 14 2020

def A074062_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) ).list()
A074062_list(60) # G. C. Greubel, Jul 05 2021

a(n) = -a(n-1) -a(n-2) -a(n-3) -a(n-4) +a(n-5), a(0)=5, a(1)=-1, a(2)=-1, a(3)=-1, a(4)=-1.

G.f.: (5 +4*x +3*x^2 +2*x^3 +x^4)/(1 +x +x^2 +x^3 +x^4 -x^5).

More terms from Michel Marcus, Sep 14 2020

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

A099163 Expansion of (1-2x^2)/((1-2x)*(1+x-x^2)).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 27, 49, 106, 199, 419, 804, 1663, 3237, 6618, 13003, 26383, 52156, 105299, 209001, 420586, 836991, 1680747, 3350548, 6718807, 13408957, 26864282, 53653539, 107428471, 214660524, 429638859, 858763489, 1718359018, 3435371767

Offset: 0

Paul Barry, Oct 01 2004

Counts closed walks of length n at the vertex with loop of the graph with adjacency matrix A=[0,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Binomial transform is A099164.

Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ (.5'i + .5i' + .5'ii' + .5e)*(.5j' + .5'kk' + .5'ki' + .5e) ], 1vesforseq = A000079(n+2) (Dement)

Michael De Vlieger, Table of n, a(n) for n = 0..3323
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,3,-2).

Cf. A039834.

CoefficientList[Series[(1 - 2 x^2)/((1 - 2 x) (1 + x - x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Sep 14 2022 *)

a(n)=sum(k=-n\5,n\5,binomial(n,(n-5*k)\2)) \\ Paul D. Hanna, Jan 02 2009

a(n)=-fibonacci(n+1)+2*sum(k=-n\10,n\10,binomial(n,n\2-5*k)) \\ Paul D. Hanna, Jan 02 2009

a(n)=a(n-1)+3a(n-2)-2a(n-3); a(n)=((sqrt(5)-1)/2)^n(3/10+sqrt(5)/10)+((-sqrt(5)-1)/2)^n(3/10-sqrt(5)/10)+2^(n+1)/5; a(n)=sum{k=0..n, (-1)^(n-k)Fib(n-k+1)(2^(k-1)+0^k/2-sum{j=0..k, C(k, j)j(-1)^j})}.
4*a(n+1) - a(n+3) = A039834(n) - Creighton Dement, Feb 25 2005
Contribution from Paul D. Hanna, Jan 02 2009: (Start)
a(n) = Sum_{k=-[n/5]..[n/5]} C(n, [(n-5*k)/2]).
a(n) = 2*Sum_{k=-[n/10]..[n/10]} C(n, [n/2]-5*k) - fibonacci(n+1). (End)
5*a(n) = 2^(n+1) + A061084(n+1), n>0. - R. J. Mathar, Sep 11 2019

cp's OEIS Frontend

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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A075091 Sum of Lucas numbers and reflected Lucas numbers (comment to A061084).

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A075151 a(n)=L(n)^2*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (comment to A061084).

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A124844 Triangle T(n,k)=binomial(n,k)*A061084(k), 0<=k<=n, read by rows.

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A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

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A203976 a(n) = 3*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=4.

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A099163 Expansion of (1-2x^2)/((1-2x)*(1+x-x^2)).