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

%I A272690 #52 Oct 19 2024 15:57:32
%S A272690 1,24,1060,48672,2238736,102981504,4737148480,217908828672,
%T A272690 10023806116096,461095081334784,21210373741388800,975677192103862272,
%U A272690 44881150836777619456,2064532938491770404864,94968515170621438443520,4368551697848586168041472,200953378101034963729186816
%N A272690 a(n) = 22*Sum_{i=0..n-2} 46^i*2^(n-2-i) + 2^(n-1).
%C A272690 This sequence gives a lower bound on the number of ways of combining n 2 X 4 LEGO blocks.
%C A272690 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 A272690 Seiichi Manyama, <a href="/A272690/b272690.txt">Table of n, a(n) for n = 1..602</a>
%H A272690 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 A272690 Jørgen Kirk Kristiansen, <a href="/A272690/a272690.jpg">Taljonglering med klodser - eller talrige klodser</a>, Klodshans 1974 [In Danish].
%H A272690 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 A272690 <a href="/wiki/Index_to_OEIS:_Section_Lc#LEGO">Index entry for sequences related to LEGO blocks</a>
%H A272690 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (48,-92).
%F A272690 From _Colin Barker_, May 31 2016: (Start)
%F A272690 a(n) = 2^(n-2)*(23+23^n)/23.
%F A272690 a(n) = 48*a(n-1) - 92*a(n-2) for n > 2.
%F A272690 G.f.: x*(1-24*x) / ((1-2*x)*(1-46*x)).
%F A272690 (End)
%F A272690 First formula follows by simplifying the formula in the definition, and the other two follow immediately. - _Rick L. Shepherd_, Jun 02 2016
%F A272690 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 A272690 t1:=n->22*add(46^i*2^(n-2-i),i=0..n-2)+2^(n-1);
%p A272690 t2:=[seq(t1(n),n=1..20)];
%t A272690 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 A272690 (Ruby)
%o A272690 def A272690(n)
%o A272690   22 * (0..n - 2).inject(0){|s, i| s + 46 ** i * 2 ** (n - 2 - i)} + 2 ** (n - 1)
%o A272690 end # _Seiichi Manyama_, May 31 2016
%o A272690 (PARI) A272690(n) = 2^(n - 2)*(1 + 23^(n - 1)) \\ _Rick L. Shepherd_, Jun 02 2016
%Y A272690 Cf. A112389, A112390.
%K A272690 nonn
%O A272690 1,2
%A A272690 _N. J. A. Sloane_, May 31 2016