A087131 -id:A087131 - cp's OEIS frontend

A052899 Expansion of g.f.: (1-2x) / ((x-1)(4x^2+2x-1)).

1, 1, 5, 13, 45, 141, 461, 1485, 4813, 15565, 50381, 163021, 527565, 1707213, 5524685, 17878221, 57855181, 187223245, 605867213, 1960627405, 6344723661, 20531956941, 66442808525, 215013444813, 695798123725, 2251650026701, 7286492548301, 23579585203405, 76305140600013

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

From L. Edson Jeffery, Apr 19 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A = A_(10,4) =
  (0 0 0 0 1)
  (0 0 0 2 0)
  (0 0 2 0 1)
  (0 2 0 2 0)
  (2 0 2 0 1).
Then a(n) = (1/5)*trace(A^n). (End)
a(n-1)+1 is the number of paths to reach a position outside a 4 X 4 chessboard after n steps, starting in one of the corners, when performing a walk with unit steps on the square lattice. - Ruediger Jehn, Oct 10 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 875
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A084057.

[(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

spec := [S,{S=Sequence(Prod(Union(Sequence(Union(Z,Z)),Z,Z),Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(1-2x)/((x-1)(4x^2+2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,-4},{1,1,5},40] (* Harvey P. Dale, Jul 10 2017 *)

makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3),x,0,n),x,n),n,0,25); /* Bruno Berselli, May 30 2011 */

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,1,2,4, lambda n:-1)
[next(it) for i in range(1,28)] # Zerinvary Lajos, Jul 09 2008

Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 2*a(n+1) + a(n+2) + 1 = 0}.
a(n) = Sum((-1/25)*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3)).
a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - Gary W. Adamson, Mar 01 2008
a(n) = (1/5) * Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - L. Edson Jeffery, Apr 19 2011
a(n) = (2*A087131(n)+1)/5. - Bruno Berselli, Apr 20 2011
a(n) = (2/5)*((1+sqrt(5))^n + (1-sqrt(5))^n + 1/2). - Ruediger Jehn, Sep 29 2024
E.g.f.: exp(x)*(1 + 4*cosh(sqrt(5)*x))/5. - Stefano Spezia, Oct 02 2024

More terms from James Sellers, Jun 08 2000

A127220 a(n) = 3^ntetranacci(n) or (2^n)A001648(n).

Original entry on oeis.org

3, 27, 189, 1215, 6318, 37179, 216513, 1253151, 7223661, 41806692, 241805655, 1398221271, 8084811933, 46753521975, 270362105694, 1563413859999, 9040715391141, 52279683047127, 302316992442837, 1748203962973380, 10109314209860523, 58458991419115875

Offset: 1

Artur Jasinski, Jan 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27,81).

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127221, A127222.

I:=[3, 27, 189, 1215]; [n le 4 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Table[Tr[MatrixPower[3*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81}, {3, 27, 189, 1215}, 50] (* G. C. Greubel, Dec 19 2017 *)

x='x+O('x^30); Vec(-3*x*(108*x^3 +27*x^2 +6*x +1)/(81*x^4 +27*x^3 +9*x^2 +3*x -1)) \\ G. C. Greubel, Dec 19 2017

a(n) = Trace of matrix [({3,3,3,3},{3,0,0,0},{0,3,0,0},{0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3) + 81*a(n-4).
G.f.: -3*x*(108*x^3+27*x^2+6*x+1)/(81*x^4+27*x^3+9*x^2+3*x-1). (End)

More terms from Colin Barker, Sep 02 2013

A127221 a(n) = 2^npentanacci(n) or (2^n)A023424(n-1).

Original entry on oeis.org

2, 12, 56, 240, 992, 3648, 14464, 57088, 224768, 883712, 3471360, 13651968, 53682176, 211075072, 829915136, 3263102976, 12830244864, 50447253504, 198353354752, 779904614400, 3066503888896, 12057176965120, 47407572189184, 186401664532480, 732912043425792

Offset: 1

Artur Jasinski, Jan 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,8,16,32).

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127222.

Cf. A023424, A074048.

I:=[2, 12, 56, 240, 992]; [n le 5 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4) + 32*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Table[Tr[MatrixPower[2*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16, 32}, {2, 12, 56, 240, 992}, 50] (* G. C. Greubel, Dec 19 2017 *)

x='x+O('x^30); Vec(-2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5)) \\ G. C. Greubel, Dec 19 2017

a(n) = Trace of matrix [({2,2,2,2,2},{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009
a(n) = 2*a(n-1)+4*a(n-2)+8*a(n-3)+16*a(n-4)+32*a(n-5). - Colin Barker, Sep 02 2013

Definition corrected by R. J. Mathar, Sep 17 2009

More terms from Colin Barker, Sep 02 2013

A127222 a(n) = 3^npentanacci(n) or (3^n)A023424(n-1).

Original entry on oeis.org

3, 27, 189, 1215, 7533, 41553, 247131, 1463103, 8640837, 50959287, 300264165, 1771292853, 10447598619, 61618989627, 363414767589, 2143339285311, 12641143135581, 74555586323649, 439717218548643, 2593383067853775, 15295369041550269, 90209719910309895

Offset: 1

Artur Jasinski, Jan 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27,81,243).

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127221.

Cf. A023424, A074048.

I:=[3, 27, 189, 1215, 7533]; [n le 5 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4) + 243*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Table[Tr[MatrixPower[3*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81, 243}, {3, 27, 189, 1215, 7533}, 50] (* G. C. Greubel, Dec 19 2017 *)

x='x+O('x^30); Vec(-3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5)) \\ G. C. Greubel, Dec 19 2017

a(n) = Trace of matrix [({3,3,3,3,3},{3,0,0,0,0},{0,3,0,0,0},{0,0,3,0,0},{0,0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 3*a(n-1)+9*a(n-2)+27*a(n-3)+81*a(n-4)+243*a(n-5). - Colin Barker, Sep 02 2013

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013

A269992 Decimal expansion of Sum_{n>=1} 2^(1-n)/L(n), where L = A000032 (Lucas numbers).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 12 2016

			1.2552211343298486031402667274403360...

Cf. A000032, A084057, A087131, A269991.

x = N[Sum[2^(1 - n)/LucasL[n], {n, 1, 500}], 100]
RealDigits[x][[1]]

L(n) = real((2 + quadgen(5)) * quadgen(5)^n); \\ A000032
suminf(n=1, 2^(1-n)/L(n)) \\ Michel Marcus, Nov 17 2020

Equals Sum_{n>=1} 1/A084057(n) = 2 * Sum_{n>=1} 1/A087131(n). - Amiram Eldar, Feb 01 2021

A272263 a(n) = numerator of A000032(n) - 1/2^n.

Original entry on oeis.org

1, 1, 11, 31, 111, 351, 1151, 3711, 12031, 38911, 125951, 407551, 1318911, 4268031, 13811711, 44695551, 144637951, 468058111, 1514668031, 4901568511, 15861809151, 51329892351, 166107021311, 537533612031, 1739495309311, 5629125066751, 18216231370751

Offset: 0

Paul Curtz, Apr 24 2016

A000032(n), Lucas numbers, and 1/2^n are autosequences of the second kind.
Then a(n)/2^n is also an autosequence of the second kind.
Their corresponding autosequences of the first kind are A000045(n) and n/2^n, the Oresme numbers.
Difference table of A000032(n) - 1/2^n:
1, 1/2,  11/4,   31/8,  111/16,    351/32,  1151/64, ...
   9/4,   9/8,  49/16,  129/32,    449/64, 1409/128, ...
        31/16,  31/32,  191/64,   511/128, 1791/256, ...
               129/64, 129/128,   769/256, ...
                       511/256,   511/256, ...
                                2049/1024, ... .
The first upper diagonal is A140323(n)/A004171(n). The main diagonal is the double, i.e. A140323(n)/A000302(n). The inverse binomial transform is the signed sequence.
Quintisections from a(2):
     11,      31,      111,      351,      1151,
   3711,   12031,    38911,   125951,    407551,
1318911, 4268031, 13811711, 44695551, 144637951,
etc.

			Numerators of a(0) =2-1=1, a(1)=1-1/2=1/2, a(2)=3-1/4=11/4, a(3)=4-1/8=31/8, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A000012, A000032, A000045, A000079, A000302, A004171, A085449, A087131, A140323.

CoefficientList[Series[(1 - 2*x + 6*x^2)/((1 - x)*(1 - 2*x - 4*x^2)), {x, 0, 30}], x] (* Robert Price, Apr 24 2016 *)
Table[Numerator[LucasL@ n - 1/2^n], {n, 0, 26}] (* Michael De Vlieger, Apr 24 2016 *)

Vec((1-2*x+6*x^2)/((1-x)*(1-2*x-4*x^2)) + O(x^50)) \\ Colin Barker, Apr 24 2016

a(n) = a(n-1) + 10*A085449(n), for n>0, a(0)=1.
a(n) = A087131(n) - 1.
From Colin Barker, Apr 24 2016: (Start)
a(n) = (-1+(1-sqrt(5))^n+(1+sqrt(5))^n).
a(n) = 3*a(n-1)+2*a(n-2)-4*a(n-3) for n>2.
G.f.: (1-2*x+6*x^2) / ((1-x)*(1-2*x-4*x^2)).
(End)

A097632 a(n) = 2^n * Lucas(n) * (n-1)!.

Original entry on oeis.org

2, 12, 64, 672, 8448, 138240, 2672640, 60641280, 1568931840, 45705461760, 1478924697600, 52646746521600, 2044394156851200, 86005817907609600, 3896481847600742400, 189139342470414336000, 9793081532749971456000, 538748376721309827072000, 31381673358053118836736000

Offset: 1

Ralf Stephan, Aug 17 2004

nonn

Number of possible well-colored cycles on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

A.H.M. Smeets, Table of n, a(n) for n = 1..100
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Cf. A000079, A000142, A000204, A066318, A087131.

a[n_] := 2^n*LucasL[n,1]*(n-1)!; Array[a,19] (* or *)
nmax=19; CoefficientList[Series[-Log[1-2x-4x^2], {x,0,nmax}], x]Range[0,nmax]! (* Stefano Spezia, Jan 15 2024 *)

def A097632(n):
    L0, L1, F, i = 1, 2, 2, 1
    while i < n:
        L0, L1, F, i = L0+L1, L0, 2*i*F, i+1
    return L0*F # A.H.M. Smeets, Jan 15 2024

E.g.f.: -log(1-2*x-4*x^2).
a(n) = A000204(n) * A066318(n).
a(n) ~ sqrt(2*Pi/n)*(2*n*phi/e)^n. - Stefano Spezia, Jan 16 2024

Definition corrected by and a(18)-a(19) from Stefano Spezia, Jan 15 2024

A270445 Expansion of 2x(1+4x) / (1-12x+16*x^2).

Original entry on oeis.org

2, 32, 352, 3712, 38912, 407552, 4268032, 44695552, 468058112, 4901568512, 51329892352, 537533612032, 5629125066752, 58948963008512, 617321555034112, 6464675252273152, 67698958146732032, 708952693724413952, 7424248994345254912

Offset: 1

Altug Alkan, Mar 17 2016

If p is an odd prime, a((p+1)/2) == 2 mod p. In other words, a((p+1)/2) - 2^p is divisible by p where p is an odd prime.

			a(2) = 32 because (1 + sqrt(5))^3 + (1 - sqrt(5))^3 = 32.

Colin Barker, Table of n, a(n) for n = 1..950
Index entries for linear recurrences with constant coefficients, signature (12,-16).

Cf. A077444, A087131.

Vec(2*x*(1+4*x)/(1-12*x+16*x^2) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(n) = 12*a(n-1) - 16*a(n-2) for n>2. G.f.: 2*x*(1+4*x) / (1-12*x+16*x^2). - Colin Barker, Mar 17 2016

a(n) = (1+sqrt(5))^(2*n-1) + (1-sqrt(5))^(2*n-1).

cp's OEIS Frontend

A052899 Expansion of g.f.: (1-2*x) / ((x-1)*(4*x^2+2*x-1)).

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A127220 a(n) = 3^n*tetranacci(n) or (2^n)*A001648(n).

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A127221 a(n) = 2^n*pentanacci(n) or (2^n)*A023424(n-1).

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A127222 a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).

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A269992 Decimal expansion of Sum_{n>=1} 2^(1-n)/L(n), where L = A000032 (Lucas numbers).

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A272263 a(n) = numerator of A000032(n) - 1/2^n.

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A097632 a(n) = 2^n * Lucas(n) * (n-1)!.

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A052899 Expansion of g.f.: (1-2x) / ((x-1)(4x^2+2x-1)).

A127220 a(n) = 3^ntetranacci(n) or (2^n)A001648(n).

A127221 a(n) = 2^npentanacci(n) or (2^n)A023424(n-1).

A127222 a(n) = 3^npentanacci(n) or (3^n)A023424(n-1).

A270445 Expansion of 2x(1+4x) / (1-12x+16*x^2).