A008998 -id:A008998 - cp's OEIS frontend

A335274 a(n) = 2*a(n-1) + a(n-3), where a(0) = 0, a(1) = 1, a(2) = 4.

0, 1, 4, 8, 17, 38, 84, 185, 408, 900, 1985, 4378, 9656, 21297, 46972, 103600, 228497, 503966, 1111532, 2451561, 5407088, 11925708, 26302977, 58013042, 127951792, 282206561, 622426164, 1372804120, 3027814801, 6678055766, 14728915652, 32485646105, 71649347976

Offset: 0

Michael Tulskikh, May 30 2020

a(n) is the number of ways to tile a 2 x n strip, with a bent tromino added to the top, with dominos and L-shaped trominos:
   _
  ||
  |||_  
  |||_||| . . .
  |||_||| . . .

			a(2) = 4 as shown by these four tilings:
   _         _         _         _
  |X|_      | |_      |X|_      | |_
  |X|X|  ,  |_|X|  ,  |X|X|  ,  |_| |
  |_ _|     |X X|     | | |     |X|_|
  |_ _|     |_ _|     |_|_|     |X X|

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A008998, A335242.

LinearRecurrence[{2, 0, 1}, {0, 1, 4}, 50] (* Paolo Xausa, Mar 20 2025 *)

concat(0, Vec(x*(1 + 2*x) / (1 - 2*x - x^3) + O(x^35))) \\ Colin Barker, Jun 04 2020

a(n) = 2*a(n-1) + a(n-3).
a(n) = 2*A008998(n-1) - A008998(n-4).
a(n) = A008998(n-1) + 2*A008998(n-2).
G.f.: x*(1 + 2*x) / (1 - 2*x - x^3). - Colin Barker, Jun 04 2020

A052599 Expansion of e.g.f.: 1/(1-2x-x^3).

Original entry on oeis.org

1, 2, 8, 54, 480, 5280, 69840, 1078560, 19031040, 377758080, 8331724800, 202138675200, 5349968870400, 153396430387200, 4736570917478400, 156702542540544000, 5529893367398400000, 207341583834857472000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 544

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

With[{nn=20},CoefficientList[Series[1/(1-2x-x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 10 2022 *)

E.g.f.: -1/(-1+2*x+x^3)
Recurrence: {a(0)=1, a(1)=2, a(2)=8, (-11*n-6-n^3-6*n^2)*a(n) +(-2*n-6)*a(n+2) +a(n+3)=0}
Sum(1/59*(16+12*_alpha^2+9*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z+_Z^3))*n!
a(n) = n!*A008998(n). - R. J. Mathar, Nov 27 2011

Definition clarified by Harvey P. Dale, Apr 10 2022

A052639 E.g.f. (1-2x)/(1-2x-x^3).

Original entry on oeis.org

1, 0, 0, 6, 48, 480, 6480, 100800, 1774080, 35199360, 776563200, 18840729600, 498640665600, 14297239756800, 441470866636800, 14605415016192000, 515412006100992000, 19325209343311872000, 767215648278503424000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 585

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

E.g.f.: (-1+2*x)/(-1+2*x+x^3)
Recurrence: {a(1)=0, a(0)=1, a(2)=0, (-11*n-6-n^3-6*n^2)*a(n)+(-2*n-6)*a(n+2)+a(n+3)=0}
Sum(-1/59*(8+6*_alpha^2-25*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z+_Z^3))*n!
a(n)= n!*A008998(n-3), n>2. - R. J. Mathar, Nov 27 2011

A100691 Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.

Original entry on oeis.org

1, 4, 12, 30, 70, 158, 352, 780, 1724, 3806, 8398, 18526, 40864, 90132, 198796, 438462, 967062, 2132926, 4704320, 10375708, 22884348, 50473022, 111321758, 245527870, 541528768, 1194379300, 2634286476, 5810101726, 12814582758

Offset: 0

Emeric Deutsch, Dec 07 2004

J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.

Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-1).

g:=series((1+z^2)*(1+z+z^2)/(1-z)/(1-2*z-z^3),z=0,35): 1,seq(coeff(g,z^n), n=1..34);

G.f.: (1+z^2)(1+z+z^2)/[(1-z)(1-2z-z^3)]= 1+2*(2+z^2)/((z-1)*(z^2+2*z-1)).
a(n) = 2*a(n-1) + a(n-3) + 6 for n >= 4.
a(n) = A008998(n+2) - A052980(n+1) - 3. - Ralf Stephan, May 15 2007
Conjecture: a(n) = A193641(n+2)-3, n>0 - R. J. Mathar, Jul 22 2022

A332491 a(n) = 2*a(n-1) + a(n-3), where a(0) = 3, a(1) = 1, a(2) = 2.

Original entry on oeis.org

3, 1, 2, 7, 15, 32, 71, 157, 346, 763, 1683, 3712, 8187, 18057, 39826, 87839, 193735, 427296, 942431, 2078597, 4584490, 10111411, 22301419, 49187328, 108486067, 239273553, 527734434, 1163954935, 2567183423, 5662101280, 12488157495, 27543498413

Offset: 0

Michael Tulskikh, Feb 13 2020

Index entries for linear recurrences with constant coefficients, signature (2,0,1).

LinearRecurrence[{2,0,1},{3,1,2},40] (* Harvey P. Dale, Apr 20 2025 *)

a(n) = 2*a(n-1) + a(n-3).
a(n) = A052980(n) + 2*A008998(n-3).
a(n) = A008998(n-1) + 3*A008998(n-3).
G.f.: (5x-3)/(x^3+2*x-1).

cp's OEIS Frontend

A335274 a(n) = 2*a(n-1) + a(n-3), where a(0) = 0, a(1) = 1, a(2) = 4.

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A052599 Expansion of e.g.f.: 1/(1-2x-x^3).

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A052639 E.g.f. (1-2x)/(1-2x-x^3).

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A100691 Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.

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A332491 a(n) = 2*a(n-1) + a(n-3), where a(0) = 3, a(1) = 1, a(2) = 2.

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