A122117 -id:A122117 - cp's OEIS frontend

A015441 Generalized Fibonacci numbers.

0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, 35839, 105469, 320503, 953317, 2876335, 8596237, 25854247, 77431669, 232557151, 697147165, 2092490071, 6275373061, 18830313487, 56482551853, 169464432775, 508359743893, 1525146340543, 4575304803901, 13726182847159

Offset: 0

Olivier Gérard

a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y) = xF(n-1)(x,y) + yF(n-2)(x,y), F(0)(x,y)=0, F(1)(x,y)=1, when y=6x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 06 2002
For n>=1: number of length-(n-1) words with letters {0,1,2,3,4,5,6,7} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Starting with offset 1 and convolved with (1, 3, 3, 3, ...) = A003462: (1, 4, 13, 40, ...). - Gary W. Adamson, May 28 2009
a(n) is identical to its inverse binomial transform signed. Differences: A102901. - Paul Curtz, Feb 23 2010
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 7*a(n-2) equals the number of 7-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 1, 1, 2, 20, 1, 6, 2, 3, 20, 5, 2, 12, 6, 20, 4, 16, 3, 18, 20, ... - R. J. Mathar, Aug 10 2012
A015441 and A015518 are the only integer sequences (from the family of homogeneous linear recurrence relation of order 2 with positive integer coefficients with initial values a(0)=0 and a(1)=1) whose ratio a(n+1)/a(n) converges to 3 as n approaches infinity. - Felix P. Muga II, Mar 14 2014
This is an autosequence of the first kind: the array of successive differences shows a main diagonal of zeros and the inverse binomial transform is identical to the sequence (with alternating signs). - Pointed out by Paul Curtz, Dec 05 2016
First two upper diagonals: A000400(n).
This is a variation on the Starhex honeycomb configuration A332243, see illustration in links. It is an alternating pattern of the 2nd iteration of the centered hexagonal numbers A003215 and centered 12-gonal 'Star' numbers A003154. - John Elias, Oct 06 2021

			G.f. = x + x^2 + 7*x^3 + 13*x^4 + 55*x^5 + 133*x^6 + 463*x^7 + 1261*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p. 317-318
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
John Elias, Illustration: A Starhex honeycomb variation
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 48.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
Index entries for linear recurrences with constant coefficients, signature (1,6).

Cf. A000079, A003462, A016153, A109466, A122117. Cf. A001656, A087451. Cf. A000400.

I:=[0,1]; [n le 2 select I[n] else Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

A015441:=n->(1/5)*((3^n)-((-2)^n)); seq(A015441(n), n=0..30); # Wesley Ivan Hurt, Mar 14 2014

a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]], {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1,6},{0,1},30] (* Harvey P. Dale, Apr 26 2011 *)
CoefficientList[Series[x/((1 + 2 x) (1 - 3 x)), {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)

PARI
```
{a(n) = (3^n - (-2)^n) / 5};
```

[lucas_number1(n,1,-6) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009

G.f.: x/((1+2*x)*(1-3*x)).
a(n) = a(n-1) + 6*a(n-2).
a(n) = (1/5)*((3^n)-((-2)^n)). - henryk.wicke(AT)stud.uni-hannover.de
E.g.f.: (exp(3*x) - exp(-2*x))/5. - Paul Barry, Apr 20 2003
a(n+1) = Sum_{k=0..ceiling(n/2)} 6^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(n) = (A000244(n) - A001045(n+1)(-1)^n - A001045(n)(-1)^n)/5. - Paul Barry, Apr 27 2004
The binomial transform of [1,1,7,13,55,133,463,...] is A122117. - Philippe Deléham, Oct 19 2006
a(n+1) = Sum_{k=0..n} A109466(n,k)*(-6)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = 3a(n-1) + (-1)^(n+1)*A000079(n-1). - Paul Curtz, Feb 23 2010
G.f.: Q(0) -1, where Q(k) = 1 + 6*x^2 + (k+2)*x - x*(k+1 + 6*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n) = (Sum_{1<=k<=n, k odd} binomial(n,k)*5^(k-1))/2^(n-1). - Vladimir Shevelev, Feb 05 2014
a(-n) = -(-1)^n * a(n) / 6^n for all n in Z. - Michael Somos, Mar 18 2014
From Peter Bala, Apr 01 2015: (Start)
Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} A087451(n)*x^n/n ).
For k = 0, 1, 2, ... and for n >= 1, (5^k)*a(n) | a((5^k)*n).
The expansion of exp( Sum_{n >= 1} a(5*n)/(5*a(n))*x^n/n ) has integral coefficients. Cf. A001656. (End)
From  Peter Bala, Jun 27 2025: (Start)
Sum_{n >= 1} (-6)^n/(a(n)*a(n+1)) = -2, since (-6)^n/(a(n)*a(n+1)) = (-2)^n/a(n) - (-2)^(n+1)/a(n+1) for n >= 1.
The following are examples of telescoping infinite products:
Product_{n >= 0} (1 + 6^n/a(2*n+2)) = 6, since (1 + 6^(2*n-1)/a(4*n))*(1 + 6^(2*n)/a(4*n+2)) = (6 - 4^(n+1)/b(n)) / (6 - 4^n/b(n-1)), where b(n) = (2*4^n + 3*9^n)/5 = A096951(n). Similarly,
Product_{n >= 1} (1 - 6^n/a(2*n+2)) = 3/13.
Product_{n >= 0} (1 + (-6)^n/a(2*n+2)) = 6/5.
Product_{n >= 1} (1 - (-6)^n/a(2*n+2)) = 15/13.
exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n. (End)

A005053 Expand (1-2x)/(1-5x).

Original entry on oeis.org

1, 3, 15, 75, 375, 1875, 9375, 46875, 234375, 1171875, 5859375, 29296875, 146484375, 732421875, 3662109375, 18310546875, 91552734375, 457763671875, 2288818359375, 11444091796875, 57220458984375

Offset: 0

N. J. A. Sloane

a(n) is the number of generalized compositions of n when there are 3*2^(i-1) different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

INVERTi transform of A180034: (1, 4, 22, 124, 700, ...). - Gary W. Adamson, Aug 10 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 883

Cf. A180034.

[ n eq 0 select 1 else 3*5^(n-1): n in [0..20] ]; // Klaus Brockhaus, Apr 04 2010

CoefficientList[Series[(1-2x)/(1-5x),{x,0,30}],x] (* or *) Join[{1}, NestList[5#&,3,29]]  (* Harvey P. Dale, Apr 25 2011 *)

x='x+O('x^50); Vec((1-2*x)/(1-5*x)) \\ G. C. Greubel, Sep 15 2017

Binomial transform of A122117. - Philippe Deléham, Oct 19 2006
a(0) = 1, a(n) = 3*5^(n-1) for n >= 1. - Philippe Deléham, Oct 19 2006
E.g.f.: (2 + 3*exp(5*x))/5. - Ilya Gutkovskiy, Aug 11 2016

Wrong formula deleted by Klaus Brockhaus, Apr 04 2010

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Bruno Berselli, Jan 09 2013

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Bruno Berselli, Table of n, a(n) for n = 0..200
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).
Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).
Cf. A052539, A024036.

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

G.f.: (2-3*x)/((1+x)*(1-4*x)).
a(n) = 4^n+(-1)^n.
a(n) = A086341(A047524(n)) for n>0, a(0)=2.
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019

A100088 Expansion of (1-x^2)/((1-2x)(1+x^2)).

Original entry on oeis.org

1, 2, 2, 4, 10, 20, 38, 76, 154, 308, 614, 1228, 2458, 4916, 9830, 19660, 39322, 78644, 157286, 314572, 629146, 1258292, 2516582, 5033164, 10066330, 20132660, 40265318, 80530636, 161061274, 322122548, 644245094, 1288490188, 2576980378

Offset: 0

Paul Barry, Nov 03 2004

A Chebyshev transform of A100087, under the mapping A(x) -> ((1-x^2)/(1+x^2)) * A(x/(1+x^2)).

A176742(n+2) = A084099(n+2) = period 4:repeat 0, -2, 0, 2.

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

Cf. A007910, A084099, A100087, A122117, A176742.

[n le 3 select Floor((n+2)/2) else 2*Self(n-1) - Self(n-2) +2*Self(n-3): n in [1..41]]; // G. C. Greubel, Jul 08 2022

CoefficientList[Series[(1-x^2)/((1-2x)(1+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2},{1,2,2},40] (* Harvey P. Dale, May 12 2011 *)

def A100088(n): return ((4<Chai Wah Wu, Apr 22 2025

def b(n): return (2/5)*(3*2^(2*n-1) + (-1)^n) # b=A122117
def A100088(n): return b(n/2) if (n%2==0) else 2*b((n-1)/2)
[A100088(n) for n in (0..60)]  # G. C. Greubel, Jul 08 2022

a(n) = (3*2^n + 2*cos(Pi*n/2) + 4*sin(Pi*n/2))/5.
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A100087(n-2*k)/(n-k).
a(n) = 2*a(n-1) + period 4:repeat 0, -2, 0, 2, with a(0) = 1.
a(n) = A007910(n+1) - A007910(n-1).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = (1/5)*(3*2^n + i^n*(1+(-1)^n) - 2*i^(n+1)*(1-(-1)^n)). - G. C. Greubel, Jul 08 2022
a(n) = A122117(n/2) if (n mod 2 = 0) otherwise 2*A122117((n-1)/2). - G. C. Greubel, Jul 21 2022

A122016 Riordan array(1, x(1+2x)/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 3, 15, 9, 1, 0, 3, 24, 36, 12, 1, 0, 3, 33, 90, 66, 15, 1, 0, 3, 42, 171, 228, 105, 18, 1, 0, 3, 51, 279, 579, 465, 153, 21, 1, 0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1, 0, 3, 69, 576, 2172, 3858, 3258, 1344, 276, 27, 1

Offset: 0

Philippe Deléham, Sep 24 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Rising and falling diagonals are A078010 and A122552.

			Triangle begins:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 3, 15,   9,    1;
  0, 3, 24,  36,   12,    1;
  0, 3, 33,  90,   66,   15,   1;
  0, 3, 42, 171,  228,  105,  18,   1;
  0, 3, 51, 279,  579,  465, 153,  21,  1;
  0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1;

Huyile Liang, Jinyang Zhang, and Yu Wang, Some properties of the matrix related to q-coloured coordination number, Filomat (2024) Vol. 38, No. 4, 1465-1477. See p. 1466.

Cf. Diagonals A000012, A008585, A062741, columns A000007, A122553, A122709, row sums A026150.

Cf. A078010, A084938, A102900, A117575, A122117, A122552, A147518.

T[n_,k_]:=SeriesCoefficient[(1-x)/(1-(y+1)*x-2*y*x^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 27 2023 *)

Sum_{k=0..n} T(n,k)*x^(n-k) = A026150(n), A102900(n) for x = 1, 2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1). - Philippe Deléham, Sep 25 2006
G.f.: (1-x)/(1-(y+1)*x-2*y*x^2). - Philippe Deléham, Jan 31 2012
Sum_{k=0..n} T(n,k)*x^k = A117575(n+1), A000007(n), A026150(n), A122117(n), A147518(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 31 2012

More terms from Stefano Spezia, Dec 27 2023

A147518 Expansion of (1-x)/(1-4x-6x^2).

Original entry on oeis.org

1, 3, 18, 90, 468, 2412, 12456, 64296, 331920, 1713456, 8845344, 45662112, 235720512, 1216854720, 6281741952, 32428096128, 167402836224, 864179921664, 4461136704000, 23029626345984, 118885325607936, 613719060507648

Offset: 0

Philippe Deléham, Nov 06 2008

nonn

Binomial transform of [1,2,13,44,205,...] = A002534(n+1).

a(n) is the number of compositions of n when there are 3 types of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,6).

Cf. A026150, A122117.

a:=[1,3];; for n in [3..30] do a[n]:=4*a[n-1]+6*a[n-2]; od; a; # G. C. Greubel, Jan 09 2020

I:=[1,3]; [n le 2 select I[n] else 4*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 09 2020

seq(coeff(series((1-x)/(1-4*x-6*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 09 2020

CoefficientList[Series[(1-x)/(1-4x-6x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,6},{1,3},30] (* Harvey P. Dale, Aug 22 2016 *)

my(x='x+O('x^30)); Vec((1-x)/(1-4*x-6*x^2)) \\ G. C. Greubel, Jan 09 2020

def A147518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-4*x-6*x^2) ).list()
A147518_list(30) # G. C. Greubel, Jan 09 2020

a(n) = 4*a(n-1) + 6*a(n-2) with a(0)=1, a(1)=3.
a(n) = Sum_{k=0..n} A122016(n,k)*3^k.
a(n) = ((10+sqrt(10))/20)*(2+sqrt(10))^n + ((10-sqrt(10))/20)*(2-sqrt(10))^n. - Richard Choulet, Nov 20 2008

A165622 a(n)=(-4)*a(n-1)+8 with a(0)=1.

Original entry on oeis.org

1, 4, -8, 40, -152, 616, -2456, 9832, -39320, 157288, -629144, 2516584, -10066328, 40265320, -161061272, 644245096, -2576980376, 10307921512, -41231686040, 164926744168, -659706976664, 2638827906664, -10555311626648, 42221246506600, -168884986026392

Offset: 0

Philippe Deléham, Sep 22 2009

lim_{n -> infinity} a(n)/a(n-1) = -4. [From Klaus Brockhaus, Sep 26 2009]

Cf. A122117,

NestList[8-4#&,1,30] (* Harvey P. Dale, Jul 11 2015 *)

G.f.: (1+7x)/(1+3x-4x^2). a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*3^(n-k). a(0)=1, a(1)=4, a(n)=4*a(n-2)-3*a(n-1).

a(n) = (8-3*(-4)^n)/5. [From Klaus Brockhaus and Paolo P. Lava, Sep 26 2009]

Corrected and extended by Harvey P. Dale, Jul 11 2015

A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 13, 5, 0, 0, 34, 19, 1, 0, 0, 89, 65, 8, 0, 0, 0, 233, 210, 42, 1, 0, 0, 0, 610, 654, 183, 11, 0, 0, 0, 0, 1597, 1985, 717, 74, 1, 0, 0, 0, 0, 4181, 5911, 2622, 394, 14, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 14 2011

Riordan array ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2)) .

			Triangle begins :
1
2, 0
5, 1, 0
13, 5, 0, 0
34, 19, 1, 0, 0
89, 65, 8, 0, 0, 0
233, 210, 42, 1, 0, 0, 0

Cf. A000045, A000079, A001519, A001870, A001906, A126124, A202207 (antidiagonal sums)

T(n,k) = 3*T(n-1,k) - T(n-2,k) + T(n-2,k-1).
G.f.: (1-x)/(1-3x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.
T(n,0) = A122367(n) = A000045(2n+1).

A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 18, 5, 0, 0, 54, 21, 1, 0, 0, 162, 81, 8, 0, 0, 0, 486, 297, 45, 1, 0, 0, 0, 1458, 1053, 216, 11, 0, 0, 0, 0, 4374, 3645, 945, 78, 1, 0, 0, 0, 0, 13122, 12393, 3888, 450, 14, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 14 2011

Riordan array ((1-x)/(1-3x), x^2/(1-3x)).
A skewed version of triangular array in A193723.
A202209*A007318 as infinite lower triangular matrices.

			Triangle begins:
  1
  2, 0
  6, 1, 0
  18, 5, 0, 0
  54, 21, 1, 0, 0
  162, 81, 8, 0, 0, 0
  486, 297, 45, 1, 0, 0, 0

Cf. A000244, A025192, A081038, A183188 (antidiagonal sums).

G.f.: (1-x)/(1-3*x-y*x^2).
T(n,k) = Sum_{j, j>=0} T(n-2-j,k-1)*3^j.
T(n,k) = 3*T(n-1,k) + T(n-2,k-1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively.

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A015441 Generalized Fibonacci numbers.

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A005053 Expand (1-2*x)/(1-5*x).

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A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

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A100088 Expansion of (1-x^2)/((1-2*x)*(1+x^2)).

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A122016 Riordan array(1, x*(1+2*x)/(1-x)).

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A147518 Expansion of (1-x)/(1-4*x-6*x^2).

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A005053 Expand (1-2x)/(1-5x).

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

A100088 Expansion of (1-x^2)/((1-2x)(1+x^2)).

A122016 Riordan array(1, x(1+2x)/(1-x)).

A147518 Expansion of (1-x)/(1-4x-6x^2).