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A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

%I A077909 #36 Apr 17 2023 15:24:59
%S A077909 1,0,0,-1,2,0,1,-4,4,-1,6,-12,9,-8,24,-33,26,-40,81,-92,92,-161,254,
%T A077909 -276,345,-576,784,-897,1266,-1936,2465,-3060,4468,-6337,7990,-10588,
%U A077909 15273,-20664,26568,-36449,51210,-67896,89585,-124108,170316,-225377,303278,-418532,566009,-754032,1025088
%N A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).
%C A077909 The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
%C A077909   ._____.     ._______.  ._______.
%C A077909   | ._. |     | ._. | |  | | ._. |
%C A077909   |_| |_|     |_| |_| |  | |_| |_|
%C A077909   |_. ._|  ,  | ._| ._|  |_. |_. |
%C A077909   | |_| |     | | |_| |  | |_| | |
%C A077909   |_____|     |_|_____|  |_____|_|. - _Alois P. Heinz_, Jan 03 2014
%H A077909 Alois P. Heinz, <a href="/A077909/b077909.txt">Table of n, a(n) for n = 0..1000</a>
%H A077909 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%H A077909 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1,2)
%F A077909 a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - _Johannes W. Meijer_, Aug 28 2013
%F A077909 G.f.: 1/(1 + x^3 - 2*x^4). - _Arkadiusz Wesolowski_, Nov 20 2013
%p A077909 a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
%p A077909         <<1, 0, 0, -1>>)[1, 1]:
%p A077909 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 20 2013
%t A077909 CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* _Jean-François Alcover_, Jun 08 2015, after _Arkadiusz Wesolowski_ *)
%o A077909 (PARI) Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ _Joerg Arndt_, Aug 28 2013
%Y A077909 Partial sums of A077976.
%Y A077909 Cf. A174249, A233427, A234312, A234931, A247126.
%K A077909 sign,easy
%O A077909 0,5
%A A077909 _N. J. A. Sloane_, Nov 17 2002