A007482 -id:A007482 - cp's OEIS frontend

A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].

0, 2, 6, 22, 78, 278, 990, 3526, 12558, 44726, 159294, 567334, 2020590, 7196438, 25630494, 91284358, 325114062, 1157910902, 4123960830, 14687704294, 52311034542, 186308512214, 663547605726, 2363259841606, 8416874736270, 29977143892022, 106765181148606

Offset: 1

Roger L. Bagula, May 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,2).

Cf. A028860, A100638.

a[1]:=0: a[2]:=2: for n from 3 to 25 do a[n]:=3*a[n-1]+2*a[n-2] od: seq(a[n],n=1..25);

LinearRecurrence[{3, 2}, {0, 2}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

A106434(n)=([0,1;2,3]^n)[1,1] /* M. F. Hasler, Dec 01 2008 */

a(n) = 3*a(n-1) + 2*a(n-2) for n>=3; a(1)=0, a(2)=2.
O.g.f.: 2*x^2/(1-3*x-2*x^2). - R. J. Mathar, Dec 05 2007
a(n) = 2 * A007482(n-2) for n >= 2.

Simplified definition and added cross reference. - M. F. Hasler, Dec 01 2008

Edited by N. J. A. Sloane, May 20 2006 and Dec 04 2008

A147840 a(n)=10a(n-1)-8a(n-2), a(0)=1, a(1)=8 .

Original entry on oeis.org

1, 8, 72, 656, 5984, 54592, 498048, 4543744, 41453056, 378180608, 3450181632, 31476371456, 287162261504, 2619811643392, 23900818341888, 218049690271744, 1989290355982336, 18148506037649408, 165570737528635392

Offset: 0

Philippe Deléham, Nov 14 2008

a(n) = sum_{k=0..n} 2^n*binomial(n,k)*A007482(k) = 2^n*A052913(n). - R. J. Mathar, Oct 15 2012

Index entries for linear recurrences with constant coefficients, signature (10,-8).

LinearRecurrence[{10,-8},{1,8},20] (* Harvey P. Dale, Dec 02 2021 *)

a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*7^k . G.f.: (1-2x)/(1-10x+8*x^2).
a(n)= ((17+3*sqrt(17))/34)*(5+sqrt(17))^n + ((17-3*sqrt(17))/34)*(5-sqrt(17))^n [From Richard Choulet, Nov 20 2008]
G.f.: (1-2x)/(1-10x+8x^2). - Harvey P. Dale, Dec 02 2021

A180144 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 2x^2)/(1 - 4x + x^2 + 2*x^3).

Original entry on oeis.org

1, 4, 13, 46, 163, 580, 2065, 7354, 26191, 93280, 332221, 1183222, 4214107, 15008764, 53454505, 190381042, 678052135, 2414918488, 8600859733, 30632416174, 109098967987, 388561736308, 1383883144897, 4928772907306

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.

The sequence above corresponds to just one A[5] vector with decimal value 16. This vector leads for the corner squares to A180143 and for the central square to A000012.

Index entries for linear recurrences with constant coefficients, signature (4, -1, -2).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,1,0,0,0,0]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

G.f.: (1-2*x^2)/(1 - 4*x + x^2 + 2*x^3).
a(n) = 4*a(n-1) - 1*a(n-2) - 2*a(n-3) with a(0)=1, a(1)=4 and a(2)=13.
a(n) = 1/4 + (21-6*A)*A^(-n-1)/68 + (21-6*B)*B^(-n-1)/68 with A=(-3+sqrt(17))/4 and B=(-3-sqrt(17))/4.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n)*(2)^(n+1)/((2*A007482(n) - 3*A007482(n-1)) - A007482(n-1)*sqrt(17)) for n >= 1.

A052986 Expansion of ( 1-2x ) / ( (x-1)(2x^2+3x-1) ).

Original entry on oeis.org

1, 2, 7, 24, 85, 302, 1075, 3828, 13633, 48554, 172927, 615888, 2193517, 7812326, 27824011, 99096684, 352938073, 1257007586, 4476898903, 15944711880, 56787933445, 202253224094, 720335539171, 2565513065700, 9137210275441, 32542656957722, 115902391424047

Offset: 0

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1060
Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).

I:=[1, 2, 7]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012

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

Join[{a=1,b=2},Table[c=3*b+2*a-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
LinearRecurrence[{4,-1,-2},{1,2,7},40] (* Vincenzo Librandi, Jun 23 2012 *)

a(n) = round((1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17))) \\ Colin Barker, Sep 02 2016

G.f.: (1-2*x)/(1-4*x+x^2+2*x^3).
Recurrence: {a(0)=1, a(1)=2, -2*a(n)-3*a(n+1)+a(n+2)+1=0}.
a(n) = Sum(-1/136*(-13-27*r+6*r^2)*r^(-1-n) where r=RootOf(1-4*_Z+_Z^2+2*_Z^3)).
a(n) = (1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17)). - Colin Barker, Sep 02 2016
4*a(n) = 1+3*A007482(n)-2*A007482(n-1) - R. J. Mathar, Feb 27 2019
a(n)-a(n-1) = A007483(n-1). - R. J. Mathar, Jan 09 2025

More terms from James Sellers, Jun 06 2000

A120723 Expansion of x(1+3x)(1+6x+16x^2)/((1-x)(1+2x)(1-3x-2x^2)).

Original entry on oeis.org

1, 11, 63, 247, 887, 3207, 11383, 40679, 144663, 515719, 1835831, 6540327, 23289943, 82955975, 295436919, 1052244583, 3747563927, 13347268359, 47536758199, 169305160871, 602988299991, 2147576619847, 7648703663351

Offset: 1

Roger L. Bagula, Aug 17 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-4,-4).

Cf. A007482.

I:=[1,11,63,247]; [n le 4 select I[n] else 2*Self(n-1) + 7*Self(n-2) -4*Self(n-3) -4*Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 20 2023

CoefficientList[Series[(1+3x)*(1 +6x +16x^2)/((1-x)*(1+2x)*(1-3x-2x^2)), {x, 0, 50}], x] (* Bruno Berselli, Apr 04 2012 *)
LinearRecurrence[{2,7,-4,-4}, {1,11,63,247}, 40] (* G. C. Greubel, Jul 20 2023 *)

A007482=BinaryRecurrenceSequence(3,2,1,3)
def A120723(n): return 12*int(n==0) - (1/6)*(46 - (-2)^n + 27*(A007482(n) - 5*A007482(n-1)))
[A120723(n) for n in range(41)] # G. C. Greubel, Jul 20 2023

G.f.: x*(1+3*x)*(1+6*x+16*x^2)/((1-x)*(1+2*x)*(1-3*x-2*x^2)). - Colin Barker, Apr 04 2012

a(n) = 12*[n=0] - 23/3 + (-2)^n/6 - (9/2)*(A007482(n) - 5*A007482(n- 1)). - G. C. Greubel, Jul 20 2023

Edited by N. J. A. Sloane, Jun 15 2007

Meaningful name from Joerg Arndt, Dec 26 2022

A201000 Primes in the Lucas U(3,-2) sequence.

Original entry on oeis.org

3, 11, 139, 1181629920803, 85408181065579888274696997717587

Offset: 1

R. J. Mathar, Jan 08 2013

nonn

The Lucas U(3,-2) sequence is 0, 1, 3, 11, 39, 139, 495, 1763, 6279, 22363..., A007482 with a leading zero. This sequence here contains U(3,-2,n) at indices n = 2, 3, 5, 23, 59, 107, 167, ...

Wikipedia, Lucas sequence

Cf. A007482.

A246313 G.f.: (-1+6x)/(1-3x-2*x^2).

Original entry on oeis.org

-1, 3, 7, 27, 95, 339, 1207, 4299, 15311, 54531, 194215, 691707, 2463551, 8774067, 31249303, 111296043, 396386735, 1411752291, 5028030343, 17907595611, 63778847519, 227151733779, 809012896375, 2881342156683, 10262052262799, 36548841101763, 130170627830887, 463609565696187, 1651169952750335

Offset: 0

N. J. A. Sloane, Aug 26 2014

Encountered during the analysis of a certain cellular automaton.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2).

I:=[-1,3]; [n le 2 select I[n] else 3*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 27 2014

a:= LRETools[REtoproc](a(n)=3*a(n-1)+2*a(n-2),a(n),{a(0)=-1,a(1)=3}):
seq(a(i),i=0..100); # Robert Israel, Aug 27 2014

CoefficientList[Series[(6 x - 1)/(1 - 3 x - 2 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 27 2014 *)

Vec((-1+6*x)/(1-3*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 02 2014

a(n) = 3*a(n-1) + 2*a(n-2) with a(0)=-1, a(1)=3.
a(n) = -(17+9*sqrt(17))/34*(3/2-sqrt(17)/2)^n+(-17+9*sqrt(17))/34*(3/2+sqrt(17)/2)^n. For n >= 3, a(n) = round((-17+9*sqrt(17))/34*(3/2+sqrt(17)/2)^n). - Robert Israel, Aug 27 2014
a(n) = 6*A007482(n-1)+A007482(n). - R. J. Mathar, Feb 27 2019

A099095 Riordan array (1,3+2x).

Original entry on oeis.org

1, 0, 3, 0, 2, 9, 0, 0, 12, 27, 0, 0, 4, 54, 81, 0, 0, 0, 36, 216, 243, 0, 0, 0, 8, 216, 810, 729, 0, 0, 0, 0, 96, 1080, 2916, 2187, 0, 0, 0, 0, 16, 720, 4860, 10206, 6561, 0, 0, 0, 0, 0, 240, 4320, 20412, 34992, 19683, 0, 0, 0, 0, 0, 32, 2160, 22680, 81648, 118098, 59049, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A007482. Diagonal sums are A053088. The Riordan array (1,s+tx) defines T(n,k)=binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).

			Rows begin {1}, {0,3}, {0,2,9}, {0,0,12,27}, {0,0,4,54,81},...

Eric W. Weisstein, Fermat Polynomial

Cf. A038220.

Number triangle T(n, k)=binomial(k, n-k)3^k*(2/3)^(n-k); Columns have g.f. (3x+2x^2)^k.

A132964 Convolution triangle of A006190.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 109, 126, 57, 12, 1, 360, 516, 306, 94, 15, 1, 1189, 2034, 1491, 600, 140, 18, 1, 3927, 7807, 6813, 3385, 1035, 195, 21, 1, 12970, 29382, 29737, 17568, 6630, 1638, 259, 24, 1, 42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1

Offset: 0

Philippe Deléham, Nov 24 2007

As a Riordan array, this is (1/(1-3x-x^2),x/(1-3x-x^2)).

T(n,k) is the number of words of length n over {0,1,2,3,4} having k letters 4 and avoiding runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017

			Triangle begins:
      1;
      3,      1;
     10,      6,      1;
     33,     29,      9,     1;
    109,    126,     57,    12,     1;
    360,    516,    306,    94,    15,     1;
   1189,   2034,   1491,   600,   140,    18,    1;
   3927,   7807,   6813,  3385,  1035,   195,   21,   1;
  12970,  29382,  29737, 17568,  6630,  1638,  259,  24,  1;
  42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1;
  ...

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A006190, A037027, A054456, A112906.

Sum_{k=0..n} T(n,k) = A001076(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A007482(n).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or k>n. - Philippe Deléham, Dec 08 2013

cp's OEIS Frontend

A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].

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A147840 a(n)=10*a(n-1)-8*a(n-2), a(0)=1, a(1)=8 .

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A180144 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 2*x^2)/(1 - 4*x + x^2 + 2*x^3).

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A052986 Expansion of ( 1-2*x ) / ( (x-1)*(2*x^2+3*x-1) ).

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A120723 Expansion of x*(1+3*x)*(1+6*x+16*x^2)/((1-x)*(1+2*x)*(1-3*x-2*x^2)).

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A201000 Primes in the Lucas U(3,-2) sequence.

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A246313 G.f.: (-1+6*x)/(1-3*x-2*x^2).

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A099095 Riordan array (1,3+2x).

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A147840 a(n)=10a(n-1)-8a(n-2), a(0)=1, a(1)=8 .

A180144 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 2x^2)/(1 - 4x + x^2 + 2*x^3).

A052986 Expansion of ( 1-2x ) / ( (x-1)(2x^2+3x-1) ).

A120723 Expansion of x(1+3x)(1+6x+16x^2)/((1-x)(1+2x)(1-3x-2x^2)).

A246313 G.f.: (-1+6x)/(1-3x-2*x^2).