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

%I #52 Oct 19 2024 15:57:32

%S 1,24,1060,48672,2238736,102981504,4737148480,217908828672,

%T 10023806116096,461095081334784,21210373741388800,975677192103862272,

%U 44881150836777619456,2064532938491770404864,94968515170621438443520,4368551697848586168041472,200953378101034963729186816

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

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

%C 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

%H Seiichi Manyama, <a href="/A272690/b272690.txt">Table of n, a(n) for n = 1..602</a>

%H S. Eilers, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.5.415">The LEGO counting problem</a>, Amer. Math. Monthly, 123 (May 2016), 415-426.

%H Jørgen Kirk Kristiansen, <a href="/A272690/a272690.jpg">Taljonglering med klodser - eller talrige klodser</a>, Klodshans 1974 [In Danish].

%H Fabien Pazuki, <a href="http://images-archive.math.cnrs.fr/Combinatoire-des-briques-LEGO.html">Combinatoire des briques LEGO</a>, Images des Mathématiques, CNRS, 2016. [In French]

%H <a href="/wiki/Index_to_OEIS:_Section_Lc#LEGO">Index entry for sequences related to LEGO blocks</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (48,-92).

%F From _Colin Barker_, May 31 2016: (Start)

%F a(n) = 2^(n-2)*(23+23^n)/23.

%F a(n) = 48*a(n-1) - 92*a(n-2) for n > 2.

%F G.f.: x*(1-24*x) / ((1-2*x)*(1-46*x)).

%F (End)

%F First formula follows by simplifying the formula in the definition, and the other two follow immediately. - _Rick L. Shepherd_, Jun 02 2016

%F 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

%p t1:=n->22*add(46^i*2^(n-2-i),i=0..n-2)+2^(n-1);

%p t2:=[seq(t1(n),n=1..20)];

%t 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 *)

%o (Ruby)

%o def A272690(n)

%o 22 * (0..n - 2).inject(0){|s, i| s + 46 ** i * 2 ** (n - 2 - i)} + 2 ** (n - 1)

%o end # _Seiichi Manyama_, May 31 2016

%o (PARI) A272690(n) = 2^(n - 2)*(1 + 23^(n - 1)) \\ _Rick L. Shepherd_, Jun 02 2016

%Y Cf. A112389, A112390.

%K nonn,changed

%O 1,2

%A _N. J. A. Sloane_, May 31 2016