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A031940 Length of longest legal domino snake using full set of dominoes up to [n:n].

%I A031940 #32 Sep 08 2022 08:44:50
%S A031940 1,3,6,9,15,19,28,33,45,51,66,73,91,99,120,129,153,163,190,201,231,
%T A031940 243,276,289,325,339,378,393,435,451,496,513,561,579,630,649,703,723,
%U A031940 780,801,861,883,946,969,1035,1059,1128,1153,1225,1251,1326,1353,1431,1459
%N A031940 Length of longest legal domino snake using full set of dominoes up to [n:n].
%H A031940 G. C. Greubel, <a href="/A031940/b031940.txt">Table of n, a(n) for n = 1..5000</a>
%H A031940 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H A031940 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A031940 C(n, 2) + n if n odd, C(n, 2) + n/2 + 1 if n even. - _T. D. Noe_, Nov 09 2006
%F A031940 a(n) = A204556(n+1) / (n+1). - _Reinhard Zumkeller_, Jan 18 2012
%F A031940 G.f.: -x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - _R. J. Mathar_, Aug 13 2012
%F A031940 a(n) = ((-1)^n*(2 - n) + (2 + n + 2*n^2))/4. - _G. C. Greubel_, Jun 15 2018
%e A031940 E.g., for n=4 [ 1:1 ][ 1:2 ][ 2:2 ][ 2:3 ][ 3:3 ][ 3:1 ][ 1:4 ][ 4:4 ][ 4:2 ].
%t A031940 Rest[CoefficientList[Series[x*(1 + 2*x + x^2 - x^3 + x^4)/((1 + x)^2*(1 - x)^3), {x, 0, 50}], x]] (* or *) Table[((-1)^n*(2-n) + (2+n+2*n^2))/4, {n,1, 50}] (* _G. C. Greubel_, Jun 15 2018 *)
%o A031940 (PARI) for(n=1, 60, print1(((-1)^n*(2 - n) + (2 + n + 2*n^2))/4, ", ")) \\ _G. C. Greubel_, Jun 15 2018
%o A031940 (PARI) Vec(-x*(1+2*x+x^2-x^3+x^4) / ( (1+x)^2*(x-1)^3 ) + O(x^60)) \\ _Felix Fröhlich_, Jun 18 2018
%o A031940 (Magma) [((-1)^n*(2 - n) + (2 + n + 2*n^2))/4: n in [1..60]]; // _G. C. Greubel_, Jun 15 2018
%Y A031940 Cf. A031878, A204556.
%K A031940 nonn
%O A031940 1,2
%A A031940 _Colin Mallows_
%E A031940 Corrected by _T. D. Noe_, Nov 09 2006
%E A031940 More terms from _Felix Fröhlich_, Jun 18 2018