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%I #44 Aug 23 2023 07:12:09
%S 0,0,2,7,17,33,57,90,134,190,260,345,447,567,707,868,1052,1260,1494,
%T 1755,2045,2365,2717,3102,3522,3978,4472,5005,5579,6195,6855,7560,
%U 8312,9112,9962,10863,11817,12825,13889,15010,16190,17430,18732,20097,21527
%N Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.
%H Vincenzo Librandi, <a href="/A045947/b045947.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - _Michael Somos_
%F a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - _Bruno Berselli_, Aug 29 2011
%F a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - _J. M. Bergot_, Apr 12 2018
%F E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - _Stefano Spezia_, Aug 22 2023
%t LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* _Jean-François Alcover_, Dec 12 2016 *)
%t CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* _Harvey P. Dale_, Jun 26 2021 *)
%o (PARI) a(n)=(4*n^3+2*n^2-4*n)\16
%o (Magma) [Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // _Vincenzo Librandi_, Aug 29 2011
%Y First differences of A082289.
%Y Cf. A000447, A002412, A051895.
%K nonn,easy,nice
%O 0,3
%A _R. K. Guy_
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