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%I #25 Dec 14 2023 17:02:10
%S 1,1,1,2,2,3,4,6,8,12,16,24,33,49,69,102,145,214,307,452,653,960,1393,
%T 2046,2978,4371,6376,9354,13665,20041,29307,42972,62884,92191,134974,
%U 197858,289772,424746,622198,911970,1336121,1958319,2869417,4205538,6162579,9031996,13235661,19398240
%N Antidiagonal sums of the antidiagonals of Losanitsch's triangle.
%C The Ca1 and Ca2 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 equal this sequence. - _Johannes W. Meijer_, Jul 14 2011
%C For n >= 5, a(n+1)-1 is the number of non-isomorphic snake polyominoes with n cells that can be inscribed in a rectangle of height 2. - _Christian Barrientos_ and _Sarah Minion_, Jul 29 2018
%H G. C. Greubel, <a href="/A102543/b102543.txt">Table of n, a(n) for n = 0..1000</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 25.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,1,-1,0,-1).
%F a(n) = A068927(n-1), n>3.
%F From _Johannes W. Meijer_, Jul 14 2011: (Start)
%F G.f.: (-1/2)*(1/(x^3+x-1)+(1+x+x^3)/(x^6+x^2-1))= ( 1-x^2-x^4-x^6 ) / ( (x^3+x-1)*(x^6+x^2-1) ).
%F a(n) = (A000930(n)+x(n)+x(n-1)+x(n-3))/2 with x(2*n) = A000930(n) and x(2*n+1) = 0. (End)
%p A102543 := proc(n): (A000930(n)+x(n)+x(n-1)+x(n-3))/2 end: A000930:=proc(n): sum(binomial(n-2*i, i), i=0..n/3) end: x:=proc(n): if type(n, even) then A000930(n/2) else 0 fi: end: seq(A102543(n), n=0..38); # _Johannes W. Meijer_, Jul 14 2011
%t CoefficientList[Series[(1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 27 2017 *)
%t LinearRecurrence[{1,1,0,0,-1,1,-1,0,-1},{1,1,1,2,2,3,4,6,8},50] (* _Harvey P. Dale_, Dec 14 2023 *)
%o (PARI) x='x+O('x^50); Vec((1 - x^2 - x^4 - x^6)/((x^3 + x - 1)*(x^6 + x^2 - 1))) \\ _G. C. Greubel_, Apr 27 2017
%Y Cf. A034851 A068927, A102451.
%K nonn
%O 0,4
%A _Gerald McGarvey_, Feb 24 2005
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