A112390 -id:A112390 - cp's OEIS frontend

A112389 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4.

Original entry on oeis.org

1, 24, 1560, 119580, 10166403, 915103765, 85747377755, 8274075616387, 816630819554486, 82052796578652749

Offset: 1

N. J. A. Sloane, Dec 06 2005

a(6) is often quoted as 102981500, but this is incorrect.

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

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, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
S. Eilers and M. Abrahamsen, Efficient counting of LEGO structures, March 30 2007.
Index entry for sequences related to LEGO blocks.

Cf. A112390, A272690.

Thanks to Gerald McGarvey, Christian Schroeder and Jud McCranie, who contributed to this entry.
a(8) from Søren Eilers, Oct 29 2006
a(9) from Johan Nilsson, Jan 06 2014
a(10) from Matthias Simon, Apr 06 2018

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.

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

cp's OEIS Frontend

A112389 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4.

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

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cp's OEIS Frontend

A112389 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Ruby

Formula

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