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%I M2832 #46 Dec 13 2021 10:56:26
%S 1,3,10,20,39,63,100,144,205,275,366,468,595,735,904,1088,1305,1539,
%T 1810,2100,2431,2783,3180,3600,4069,4563,5110,5684,6315,6975,7696,
%U 8448,9265,10115,11034,11988,13015,14079,15220,16400,17661,18963,20350,21780,23299
%N Number of paraffins.
%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Enrique Pérez Herrero, <a href="/A005997/b005997.txt">Table of n, a(n) for n = 1..5000</a>
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F G.f.: (x^3+3*x^2+x+1)*x / ((-1+x)^2*(-1+x^2)^2).
%F a(n) = A005999(n)+(n-1)^2. - _Enrique Pérez Herrero_, Mar 27 2012
%F a(n) = 1 + floor((n-1)/2) + 2*(C(n+1,3)-C(floor((n+1)/2),3)-C(ceiling((n+1)/2),3). - _Enrique Pérez Herrero_, Apr 22 2012
%F a(n) = (n+1)(2n^2-(-1)^n+1)/8. - _Bruno Berselli_, Apr 22 2012
%F a(n) = A004526(n) + 2*A111384(n). - _Enrique Pérez Herrero_, Apr 25 2012
%F E.g.f.: (x*(3 + 4*x + x^2)*cosh(x) + (1 + 2*x + 4*x^2 + x^3)*sinh(x))/4. - _Stefano Spezia_, Dec 13 2021
%p a:= n-> (Matrix([[0, 0, -1, -5, -12, -26]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=1..50); # _Alois P. Heinz_, Jul 31 2008
%t A005997[n_]:=1+Floor[(n-1)/2]+2*(Binomial[n+1,3]-Binomial[Floor[(n+1)/2],3]-Binomial[Ceiling[(n+1)/2],3]); Array[A005997,37] (* _Enrique Pérez Herrero_, Apr 22 2012 *)
%t LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 3, 10, 20, 39, 63}, 37] (* _Bruno Berselli_, Apr 22 2012 *)
%Y Cf. A005999.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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