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%I #32 Jan 21 2018 16:55:58
%S 2,4,14,32,82,188,438,984,2202,4852,10622,23056,49762,106796,228166,
%T 485448,1029162,2174820,4582670,9631360,20194802,42253724,88235734,
%U 183927992,382769082,795364308,1650380958,3420066544,7078742402
%N Expansion of 2*x / ((1+x)^2*(1-2*x)^2).
%C Number of 2 X 2 tiles in all tilings of a 3 X (n+1) rectangle with 1 X 1 and 2 X 2 square tiles. - _Emeric Deutsch_, Feb 18 2007
%C The terms of this sequence have a primitive divisor for all terms beyond the 4th if and only if n is not of the form 4k+2, for some nonnegative integer k. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007
%H G. C. Greubel, <a href="/A095977/b095977.txt">Table of n, a(n) for n = 1..1000</a>
%H Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>
%H A. Flatters, <a href="http://arxiv.org/abs/0708.2190">Prime divisors of some Lehmer-Pierce sequences</a>, arXiv:0708.2190 [math.NT], 2007.
%H R. P. Grimaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Grimaldi/grimaldi5.html">Tilings, Compositions, and Generalizations</a>, J. Int. Seq. 13 (2010), 10.6.5, page 7.
%H H. Prodinger, <a href="http://www.emis.de/journals/INTEGERS/papers/a8/a8.Abstract.html">On binary representations of integers with digits -1,0,1</a>, Integers 0 (2000), #A08.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-4)
%F a(n) = (1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n).
%F a(n) = 2 * A073371(n-1).
%F a(n) = Sum_{k=0..floor((n+1)/2)} k*2^k*binomial(n+1-k,k). - _Emeric Deutsch_, Feb 18 2007
%p a:=n->n/9*2^(n+2)+1/27*2^(n+3)-2*n/9*(-1)^n-8/27*(-1)^n: seq(a(n),n=1..30); # _Emeric Deutsch_, Feb 18 2007
%t Table[(1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n) , {n,1,50}] (* _G. C. Greubel_, Dec 28 2016 *)
%o (PARI) Vec(2*x / ((1+x)^2 * (1-2*x)^2) + O(x^50)) \\ _Michel Marcus_, Nov 07 2015
%Y Cf. A128099, A073371.
%K nonn,easy
%O 1,1
%A _Ralf Stephan_, Jul 16 2004
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