A112389 -id:A112389 - cp's OEIS frontend

A123762 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2.

Original entry on oeis.org

1, 4, 37, 375, 4493, 56848, 753536, 10283622, 143607345, 2041497919, 29446248496, 429858432108

Offset: 1

Søren Eilers, Oct 29 2006

M. Abrahamsen and S. Eilers, On the asymptotic enumeration of LEGO structures, Exper Math. 20 (2) (2011) 145-152.
B. Durhuus and S. Eilers, On the entropy of LEGO, arXiv:math/0504039 [math.CO], 2005.
B. Durhuus and S. Eilers, On the entropy of LEGO, J. Appl. Math. Comput. 45 (1-2) (2014), 433-448.
S. Eilers, A LEGO Counting problem, 2005.
S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
Index entry for sequences related to LEGO blocks

Cf. A007576, A082679, A112389, A112390.

A123827 Number of ways to build a contiguous building with n LEGO blocks of size 2 X 3 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.

Original entry on oeis.org

1, 13, 25, 352, 1006, 15510, 46928, 744211, 2479362

Offset: 1

Søren Eilers, Oct 29 2006

The base block is not counted among the n and must be the only block in the bottom layer of the building.

Index entry for sequences related to LEGO blocks

Cf. A112389, A123762 - A123827, A123829 - A123849.

A123829 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4 which is symmetric after a rotation by 180 degrees.

Original entry on oeis.org

1, 2, 44, 185, 3276, 15682, 282377, 1480410, 26264942

Offset: 1

Søren Eilers, Oct 29 2006

nonn

Index entry for sequences related to LEGO blocks

Cf. A112389, A123762 - A123827, A123829 - A123849.

A112390 Triangle read by rows: T(n,k) is the number of ways of building a building of height k using n 2 X 4 LEGO blocks, counted up to symmetry (for 2 <= k <= n).

Original entry on oeis.org

24, 500, 1060, 11707, 59201, 48672, 248688, 3203175, 4425804, 2238736, 7946227, 162216127, 359949655, 282010252, 102981504

Offset: 2

N. J. A. Sloane, Dec 06 2005

			Triangle begins:
       24;
      500,      1060;
    11707,     59201,     48672;
   248688,   3203175,   4425804,   2238736;
  7946227, 162216127, 359949655, 282010252, 102981504;

Anthony Lane, The Joy of Bricks, The New Yorker, Apr 27-May 04, 1998, pp. 96-103.

B. Durhuus and S. Eilers, On entropy of LEGO, arXiv:math/0504039 [math.CO], 2005.
S. Eilers, A LEGO Counting problem.
Index entry for sequences related to LEGO blocks.

Cf. A112389.

Thanks to Gerald McGarvey, Christian Schroeder and Jud McCranie, who contributed to this entry.

A272690 a(n) = 22Sum_{i=0..n-2} 46^i2^(n-2-i) + 2^(n-1).

Original entry on oeis.org

1, 24, 1060, 48672, 2238736, 102981504, 4737148480, 217908828672, 10023806116096, 461095081334784, 21210373741388800, 975677192103862272, 44881150836777619456, 2064532938491770404864, 94968515170621438443520, 4368551697848586168041472, 200953378101034963729186816

Offset: 1

N. J. A. Sloane, May 31 2016

nonn

This sequence gives a lower bound on the number of ways of combining n 2 X 4 LEGO blocks.

The formula as given was found at the LEGO Company in 1974 and the numbers a(2), a(3), a(6) were used in communication until the emergence of A112389. - Søren Eilers, Aug 02 2018

Seiichi Manyama, Table of n, a(n) for n = 1..602
S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
Jørgen Kirk Kristiansen, Taljonglering med klodser - eller talrige klodser, Klodshans 1974 [In Danish].
Fabien Pazuki, Combinatoire des briques LEGO, Images des Mathématiques, CNRS, 2016. [In French]
Index entry for sequences related to LEGO blocks
Index entries for linear recurrences with constant coefficients, signature (48,-92).

Cf. A112389, A112390.

t1:=n->22*add(46^i*2^(n-2-i),i=0..n-2)+2^(n-1);
t2:=[seq(t1(n),n=1..20)];

Table[22*Sum[46^k * 2^(n-k-2), {k,0,n-2}] + 2^(n-1), {n,1,25}] (* G. C. Greubel, May 31 2016 *)

A272690(n) = 2^(n - 2)*(1 + 23^(n - 1)) \\ Rick L. Shepherd, Jun 02 2016

def A272690(n)
  22 * (0..n - 2).inject(0){|s, i| s + 46 ** i * 2 ** (n - 2 - i)} + 2 ** (n - 1)
end # Seiichi Manyama, May 31 2016

From Colin Barker, May 31 2016: (Start)
a(n) = 2^(n-2)*(23+23^n)/23.
a(n) = 48*a(n-1) - 92*a(n-2) for n > 2.
G.f.: x*(1-24*x) / ((1-2*x)*(1-46*x)).
(End)
First formula follows by simplifying the formula in the definition, and the other two follow immediately. - Rick L. Shepherd, Jun 02 2016
Since there are 46 ways to attach one such brick on top of another, 2 of which are self-symmetric, the number of buildings with n 2 X 4 LEGO bricks of maximal height becomes a(n) = (46^(n-1) + 2^(n-1))/2 when adjusted for rotation in the XY-plane. That this is the same as the original formula found at LEGO follows by isolating a finite geometric series. - Søren Eilers, Aug 02 2018

A123824 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 3.

Original entry on oeis.org

1, 16, 697, 34958, 1947321, 114675898, 7026150623, 443114008903, 28575438072945, 1875587744568411

Offset: 1

Søren Eilers, Oct 29 2006

Matthias Simon, Program to compute a(1)...a(10)

Cf. A112389, A123762 - A123827, A123829 - A123849.

a(9) from Matthias Simon, Aug 29 2020

a(10) from Matthias Simon, Nov 15 2020

A123770 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3.

Original entry on oeis.org

1, 8, 152, 3594, 96105, 2734356, 81126946, 2480378897, 77597683206, 2472038270424

Offset: 1

Søren Eilers, Oct 29 2006

S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
Index entry for sequences related to LEGO blocks

Cf. A112389, A123762-A123827, A123829-A123849.

A123771 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is symmetric after a rotation by 180 degrees.

Original entry on oeis.org

1, 2, 12, 45, 264, 1128, 6991, 32202, 202856, 973242, 6204349

Offset: 1

Søren Eilers, Oct 29 2006

S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
Index entry for sequences related to LEGO blocks

Cf. A112389, A123762-A123827, A123829-A123849.

A123772 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is flat, i.e., with all blocks in parallel position.

Original entry on oeis.org

1, 3, 17, 107, 761, 5607, 42730, 331851, 2618024, 20895964, 168398683, 1367838390, 11184711448, 91976811135

Offset: 1

Søren Eilers, Oct 29 2006

Cf. A112389, A123762 - A123827, A123829 - A123849.

A123773 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is flat, i.e., with all blocks in parallel position and symmetric after a rotation by 180 degrees.

Original entry on oeis.org

1, 1, 3, 4, 16, 24, 103, 163, 721, 1173, 5283, 8783, 39912, 67381, 308314, 526553, 2421040, 4172834, 19257309, 33438244, 154761065, 270372085

Offset: 1

Søren Eilers, Oct 29 2006

Cf. A112389, A123762 - A123827, A123829 - A123849.

More terms by Søren Eilers, Sep 12 2018

cp's OEIS Frontend

A123762 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2.

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A123827 Number of ways to build a contiguous building with n LEGO blocks of size 2 X 3 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.

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A123829 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4 which is symmetric after a rotation by 180 degrees.

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A112390 Triangle read by rows: T(n,k) is the number of ways of building a building of height k using n 2 X 4 LEGO blocks, counted up to symmetry (for 2 <= k <= n).

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A272690 a(n) = 22*Sum_{i=0..n-2} 46^i*2^(n-2-i) + 2^(n-1).

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A123824 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 3.

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A123770 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3.

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A123771 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is symmetric after a rotation by 180 degrees.

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A123772 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is flat, i.e., with all blocks in parallel position.

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A123773 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 3 which is flat, i.e., with all blocks in parallel position and symmetric after a rotation by 180 degrees.

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Crossrefs

Extensions

A272690 a(n) = 22Sum_{i=0..n-2} 46^i2^(n-2-i) + 2^(n-1).