A006253 -id:A006253 - cp's OEIS frontend

A217233 Expansion of (1-2x+x^2)/(1-3x-3*x^2+x^3).

1, 1, 7, 23, 89, 329, 1231, 4591, 17137, 63953, 238679, 890759, 3324361, 12406681, 46302367, 172802783, 644908769, 2406832289, 8982420391, 33522849271, 125108976697, 466913057513, 1742543253359, 6503259955919, 24270496570321, 90578726325361

Offset: 0

Bruno Berselli, Sep 28 2012

Numbers with the property a(n)^2+a(n-1)^2 = 2*(a(n)-a(n-1)-(-1)^n)^2.

			a(3)=23, a(2)=7: 23^2+7^2 = 2*(23-7-(-1)^3)^2 = 578;
a(6)=1231, a(5)=329: 1231^2+329^2 = 2*(1231-329-(-1)^6)^2 = 1623602.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A109437 (1/(1-3*x-3*x^2+x^3)), A006253 ((1-x)/(1-3*x-3*x^2+x^3)).

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+x^2)/(1-3*x-3*x^2+x^3)));

CoefficientList[Series[(1 - 2 x + x^2)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 25}], x]

makelist(coeff(taylor((1-2*x+x^2)/(1-3*x-3*x^2+x^3), x, 0, n), x, n), n, 0, 25);

Vec((1-2*x+x^2)/(1-3*x-3*x^2+x^3)+O(x^26))

G.f.: (1-x)^2/((1+x)*(1-4*x+x^2)).
a(n) = (4*(-2)^n+(1-sqrt(3))^(2*n+1)+(1+sqrt(3))^(2*n+1))/(6*2^n).
a(n) = -a(-n-1) = 3*a(n-1)+3*a(n-2)-a(n-3) = 4*a(n-1)-a(n-2)+4*(-1)^n.
a(n)+a(n-1) = A052530(n) with a(-1)=-1.
a(n)-a(n-2) = A003699(n) with n>1.
Sum(a(i), i=0..n) = A006253(n).

A049507 Number of perfect matchings in graph P_{6} x P_{6} x P_{n}.

Original entry on oeis.org

1, 6728, 53786626921, 57248060375968384, 123115692449982216049513, 216388579168758145017797108072

Offset: 0

Per H. Lundow

P. H. Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report #12 (1996).
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.

Cf. A006253, A028452, A028454.

A060635 a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.

Original entry on oeis.org

2, 8, 72, 450, 3200, 21632, 149058, 1019592, 6993800, 47922050, 328499712, 2251473408, 15432082562, 105772401800, 724976569800, 4969058770242, 34058447431808, 233440040239232, 1600021920672450, 10966713178192200, 75166970919070472, 515202081704384258, 3531247605071972352

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 16 2001

nonn

The relevant graph has rotational symmetry so the number of tilings is a square or twice a square, in this case by the formula for a(n) it is always twice a square.

			a(1) = 2 because in this case the set S is the unit square and there is one horizontal tiling and one vertical.

Harry J. Smith, Table of n, a(n) for n = 1..200
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045, A001654, A006253, A004003, A006125, A002878, A049658, A197424.

with(combinat): for n from 1 to 40 do printf(`%d,`,2*fibonacci(n)^2*fibonacci(n+1)^2) od:

2*Times @@@ Partition[Fibonacci[Range[25]]^2, 2, 1] (* Paolo Xausa, Jul 03 2025 *)

{ a=1; b=0; c=1; for (n=1, 200, f=a+b; g=b+c; a=b; b=c; c=g; write("b060635.txt", n, " ", 2*f^2*g^2); ) } \\ Harry J. Smith, Jul 08 2009

a(n) = 2 * F(n)^2 * F(n+1)^2 where F(n) is the n-th Fibonacci number - sequence A000045.
G.f.: -2*x*(1-x+x^2) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - R. J. Mathar, Jan 30 2011
a(n) = -4*(-1)^n*A002878(n)/25 - 2/25 + 6*A049658(n)/25. - R. J. Mathar, Jan 30 2011
a(n) = 2 * A001654(n)^2 = 2 * A197424(n-2) for n>=2. - Alois P. Heinz, Jul 03 2025

More terms from James Sellers, Apr 16 2001

A105968 a(n) = 4a(n-1) - a(n-2) - 2(-1)^n, a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 13, 50, 185, 692, 2581, 9634, 35953, 134180, 500765, 1868882, 6974761, 26030164, 97145893, 362553410, 1353067745, 5049717572, 18845802541, 70333492594, 262488167833, 979619178740, 3655988547125, 13644335009762, 50921351491921, 190041070957924

Offset: 0

Creighton Dement, Apr 28 2005

This sequence is the (type 1A) "jbasejfor" transformation of the sequence (-1, -1, -1, -1, ..) with respect to the floretion given in the program code. Under the same conditions, the jbasejfor transformation of the sequence (1, 1, 1, 1, ...) is A006253 [Number of perfect matchings (or domino tilings) in C_4 X P_n]; the jbasejfor transformation of the sequence (1, -1, 1, -1, ...) is A001075 [Chebyshev's T(n,x) polynomials evaluated at x=2]; the jbasejfor transformation of the sequence (-1, 1, -1, 1, ...) is A001353 [3*a(n)^2 + 1 is a perfect square]. In this sense, the sequences (a(n)), A006253, A001075 and A001353 form a "quartett".

Floretion Algebra Multiplication Program, FAMP Code: 4jbasejforseq[ + .25'i + .25'j + .25'k + .25i' + .25j' + .25k' + .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' + .25e]. ForType: 1A. 1vesforseq = (-1, -1, -1, -1, ..).

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

Cf. A001075, A001353, A006253, A054491.

a:=[1,4,13];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 15 2020

I:=[1,4,13]; [n le 3 select I[n] else 3*Self(n-1) +3*Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2020

seq( simplify((4*ChebyshevU(n,2) -5*ChebyshevU(n-1,2) -(-1)^n)/3), n = 0..30); # G. C. Greubel, Jan 15 2020

Table[(4*ChebyshevU[n, 2] -5*ChebyshevU[n-1, 2] -(-1)^n)/3, {n,0,30}] (* G. C. Greubel, Jan 15 2020 *)
nxt[{n_,a_,b_}]:={n+1,b,4b-a-2(-1)^(n+1)}; NestList[nxt,{1,1,4},30][[;;,2]] (* or *) LinearRecurrence[ {3,3,-1},{1,4,13},30] (* Harvey P. Dale, Apr 03 2024 *)

Vec((1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, May 25 2015

[(4*chebyshev_U(n,2) -5*chebyshev_U(n-1,2) -(-1)^n)/3 for n in (0..30)] # G. C. Greubel, Jan 15 2020

G.f.: (1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)).
a(n) + a(n+1) = A054491(n+1) - A054491(n).
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3). - Colin Barker, May 25 2015
a(n) = ( 4*ChebyshevU(n,2) - 5*ChebyshevU(n-1,2) - (-1)^n )/3. - G. C. Greubel, Jan 15 2020
E.g.f.: (exp(2*x)*(4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - cosh(x) + sinh(x))/3. - Stefano Spezia, Sep 19 2023

A360066 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 11, 444, 13311, 422617, 13265660, 417336617, 13123557903, 412719195520, 12979269602143, 408175860119021, 12836425011761592, 403683424226081169, 12695147020245034099, 399240466722076292612, 12555423726269799691295, 394846409914451855949249

Offset: 0

Gerhard Kirchner, Jan 30 2023

nonn

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 10.

Index entries for linear recurrences with constant coefficients, signature (26,176,-146,-14,-140,27).

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065.

Maxima
```
/* See A359884. */
```

G.f.: (1 - 15*x - 18*x^2 - 23*x^3 + 7*x^4) / (1 - 26*x - 176*x^2 + 146*x^3 + 14*x^4 + 140*x^5 - 27*x^6).
Recurrence 1:
a(n) = 11*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 29*a(n-2) + 6*b(n-2) + c(n-2) + 2*d(n-2),
b(n) = 32*a(n-1) + 9*b(n-1) + 4*c(n-1) + 2*d(n-1) + e(n-1),
c(n) = 52*a(n-1) + 14*b(n-1) + 5*c(n-1) + 4*d(n-1) + 2*e(n-1),
d(n) = 14*a(n-1) + 3*b(n-1) + d(n-1),
e(n) = 48*a(n-1) + 11*b(n-1) + 2*c(n-1) + 2*d(n-1),
with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 26*a(n-1) + 176*a(n-2) - 146*a(n-3) - 14*a(n-4) - 140*a(n-5) + 27*a(n-6) for n >= 6. For n < 6, recurrence 1 can be used.

A360575 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and 2 X 2 X 1 plates.

Original entry on oeis.org

1, 8, 153, 2470, 41571, 693850, 11602579, 193942076, 3242104149, 54196828452, 905988148597, 15145052657186, 253174020910071, 4232212575080006, 70748267813548207, 1182671546039152712, 19770264765434877913, 330491902143708738464

Offset: 0

Gerhard Kirchner, Feb 12 2023

nonn

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 11.

Index entries for linear recurrences with constant coefficients, signature (16,21,-157,100,65,-42).

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360576, A360577.

G.f.: (1-8*x+4*x^2+11*x^3-6*x^4) / (1-16*x-21*x^2+157*x^3-100*x^4-65*x^5+42*x^6).
Recurrence 1:
a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)
b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)
d(n) = 2*a(n-1) + b(n-1) + d(n-1)
e(n) = 12*a(n-1) + 3*b(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=16*a(n-1) + 21*a(n-2) - 157*a(n-3) + 100*a(n-4) + 65*a(n-5) - 42*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 6, 122, 1768, 28844, 457592, 7318760, 116806896, 1865305376, 29782666544, 475549098160, 7593154541264, 121241257906000, 1935879286697296, 30910512661708432, 493553365105565264, 7880649886335326608, 125831666350680625104

Offset: 0

Gerhard Kirchner, Feb 12 2023

nonn

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 12.

Index entries for linear recurrences with constant coefficients, signature (15,28,-214,192,384,-128).

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360577.

G.f.: (1-9*x+4*x^2-16*x^3) / (1-15*x-28*x^2+214*x^3-192*x^4-384*x^5+128*x^6).
Recurrence 1:
a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)
b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)
d(n) = 2*a(n-1) + b(n-1) + d(n-1)
e(n) = 12*a(n-1) + 3*b(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=15*a(n-1) + 28*a(n-2) - 214*a(n-3) + 192*a(n-4) + 384*a(n-5) - 128*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 3, 60, 657, 8311, 101284, 1246049, 15292819, 187803572, 2305968393, 28315208039, 347681742812, 4269186204201, 52421329940803, 643681521419708, 7903765218510353, 97050331862075975, 1191681006432895780, 14632650860374551265, 179674317212728197891, 2206220907971874345652

Offset: 0

Gerhard Kirchner, Feb 12 2023

nonn

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 13.

Index entries for linear recurrences with constant coefficients, signature (8,51,27,-96,-43,66).

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360576.

G.f.: (1-5*x-15*x^2-3*x^3+10*x^4) / (1-8*x-51*x^2-27*x^3+96*x^4+43*x^5-66*x^6).
Recurrence 1:
a(n) = 3*a(n-1) + b(n-1) + c(n-1) + 19*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2)
b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1)
c(n) = 20*a(n-1) + 6*b(n-1) + 2*c(n-1) + 2*e(n-1)
d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1)
e(n) = 24*a(n-1) + 7*b(n-1) + 2*c(n-1) + 2*d(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=8*a(n-1) + 51*a(n-2) + 27*a(n-3) - 96*a(n-4) - 43*a(n-5) + 66*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

A360645 Number of 4-dimensional tilings of a 2 X 2 X 2 X n box with 2 X 2 X 1 X 1 plates.

Original entry on oeis.org

1, 3, 30, 177, 1281, 8520, 58629, 397887, 2715510, 18490533, 126023349, 858595560, 5850498441, 39863005323, 271617783150, 1850725023657, 12610357769721, 85923544106760, 585460036653789, 3989166905015367, 27181111280961990, 185204779320272253

Offset: 0

Gerhard Kirchner, Feb 15 2023

nonn

The figure shows three 2 X 2 X 2 cubes as intersections of three successive hyperplanes (distance 1) with the box. The 3-d cross-section of a 2 X 2 X 1 X 1 plate is a 2 X 2 X 1 plate (p4) as part of one cube or a 2 X 1 X 1 domino if the plate (p2) connects two cubes. p4 or p2 indicates the number of unit cubes on the current level (hyperplane). PQRS and P'Q'R'S' (not visible: P') is one of three ways to select a pair of p4-plates. Q'Q"R"S' represents a p2-plate.
Suppose the box is completely tiled up to a certain level. Then the next (current) level may be empty (profile A0) or not (profile B0). The index 0 is used for the current level and continued with 1,2... Transitions:
a) A0->3*A1 (3 ways of selecting a pair of p4-plates, also A001045(2)=3).
b) A0->9*A2 (9 ways of tiling a 2 X 2 X 2 cube with 3d-dominos, also A006253(2)=9).
c) A0->12*B1. One p4-plate and two p2-plates can be selected in 12 ways: 6 faces of the 2 X 2 X 2 cube and two ways of selecting a pair of dominos on each face. They tile the next level with corresponding dominos. A further nonempty profile does not occur. Also, A359884(2)-A006253(2)-A001045(2)=24-9-3=12.
d) B0->1*A1 (one accomplishing p4-plate is placed on B0).
e) B0->*2B1 (2 ways of selecting a pair of dominos on B0).
Let a(n) and b(n) be the number of tilings of the 2 X 2 X 2 X n box ending with an A- or a B-profile respectively. With the transitions above, one obtains recurrence 1.
         /\               /\               /\
       /    \           /    \           /    \
     / \  S' /\       / \     /\       / \     /\
   /     \ /    \   /     \ /    \   /     \ /    \
  |\     / \     /||\     / \     /||\     / \     /|
  |  \ /     \ /  ||  \ /     \ /  ||  \ /     \ /  |
  | S |\     /| R'||   |\     /| R"||   |\     /|   |---> 4th dimension
  |\  |  \ /  |  /||\  |  \ /  |  /||\  |  \ /  |  /|
  |  \| R |   |/  ||  \|   |   |/  ||  \|   |   |/  |
  | P |\  |  /| Q'||   |\  |  /| Q"||   |\  |  /|   |
   \  |  \|/  |  /  \  |  \|/  |  /  \  |  \|/  |  /
     \| Q |   |/      \|   |   |/      \|   |   |/
       \  |  /          \  |  /          \  |  /
         \|/              \|/              \|/

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,15,-18).

Cf. A001045, A006253, A359884.

LinearRecurrence[{5, 15, -18}, {1, 3, 30}, 25] (* Paolo Xausa, May 28 2024 *)

Recurrence 1: a(n) = 3*a(n-1) + b(n-1) + 9*a(n-2), b(n) = 12*a(n-1) + 2*b(n-1), with a(0) = 1 and a(-1) = b(0) = 0.
Recurrence 2: a(n) = 5*a(n-1) + 15*a(n-2) - 18*a(n-3).
G.f.: (1-2*x) / (1-5*x-15*x^2+18*x^3).

A360644 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

Original entry on oeis.org

1, 12, 513, 16194, 547543, 18234354, 609298887, 20344385080, 679408772089, 22688284005780, 757662377924917, 25301659203704234, 844933359518672599, 28216027727373068302, 942256839186226313727, 31466085716246304261600, 1050790517091131646143477

Offset: 0

Gerhard Kirchner, Feb 15 2023

nonn

Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 14.

Index entries for linear recurrences with constant coefficients, signature (28,195,-497,30,-79,66).

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065.

G.f.: (1-16*x-18*x^2-13*x^3+10*x^4) / (1-28*x-195*x^2+497*x^3-30*x^4+79*x^5-66*x^6)
Recurrence 1:
a(n) = 12*a(n-1) + 4*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 43*a(n-2) + 8*b(n-2) + c(n-2) + 2*d(n-2)
b(n) = 32*a(n-1) + 9*b(n-1) + 4*c(n-1) + 2*d(n-1) + e(n-1)
c(n) = 60*a(n-1) + 16*b(n-1) + 6*c(n-1) + 4*d(n-1) + 2*e(n-1)
d(n) = 14*a(n-1) + 3*b(n-1) + d(n-1)
e(n) = 64*a(n-1) + 13*b(n-1) + 2*c(n-1) + 2*d(n-1)
with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
Recurrence 2:
a(n)=28*a(n-1) + 195*a(n-2) - 497*a(n-3) + 30*a(n-4) - 79*a(n-5) + 66*a(n-6)
for n>=6. For n<6, recurrence 1 can be used.

cp's OEIS Frontend

A217233 Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).

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A049507 Number of perfect matchings in graph P_{6} x P_{6} x P_{n}.

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A060635 a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.

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A105968 a(n) = 4*a(n-1) - a(n-2) - 2*(-1)^n, a(0) = 1, a(1) = 4.

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Magma

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A360066 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

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Maxima

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A360575 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 1 X 1 dominos and 2 X 2 X 1 plates.

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A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

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A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

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A217233 Expansion of (1-2x+x^2)/(1-3x-3*x^2+x^3).

A105968 a(n) = 4a(n-1) - a(n-2) - 2(-1)^n, a(0) = 1, a(1) = 4.