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%I #20 Mar 07 2024 16:26:59
%S 1,1,3,11,35,107,339,1073,3375,10633,33525,105651,332941,1049305,
%T 3306957,10421967,32845327,103513709,326228241,1028123557,3240180157,
%U 10211580633,32182277499,101423965833,319642412979,1007368140211,3174768208785,10005431759263
%N "Model 1" for number of free alkanes on n points.
%C Linear recurrence and empirical g.f confirmed for more terms. - _Ray Chandler_, Mar 07 2024
%D Gy. Tasi et al., Quantum algebraic-combinatoric study of the conformational properties of n-alkanes II, J. Math. Chemistry, 27 (2000), 191-199 (see Table 1).
%H Ray Chandler, <a href="/A126939/b126939.txt">Table of n, a(n) for n = 0..2006</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, 7, 3, 3, 1).
%F Define sequences a[n], b[n], c[n], d[n] by the recurrences shown in the Maple code below. Sequence gives values of a[n] and also (with a different offset) a[n]+b[n]+d[n].
%F Empirical g.f.: (x^5+3*x^3+x-1) / (x^6+3*x^5+3*x^4+7*x^3+x^2+2*x-1). - _Colin Barker_, Apr 08 2013
%p M:=35; a:=array(-5..M); b:=array(-5..M); c:=array(-5..M); d:=array(-5..M);
%p for i from -5 to 0 do a[i]:=0; b[i]:=0; c[i]:=0; d[i]:=0; od: a[0]:=1;
%p for n from 1 to M do
%p a[n]:=a[n-1]+b[n-1]+d[n-1];
%p b[n]:=2*a[n-1]+b[n-1]+b[n-3]+c[n-3]+c[n-4];
%p c[n]:=2*a[n-1]+b[n-1]+b[n-2]+b[n-3]+2*c[n-3]+c[n-4];
%p d[n]:=b[n-1]+b[n-2]+c[n-1]+2*c[n-2]+c[n-3]; od:
%t a[0] = a[1] = 1; a[2] = 3; a[3] = 11; a[4] = 35; a[5] = 107; a[n_] := a[n] = a[n-6] + 3*a[n-5] + 3*a[n-4] + 7*a[n-3] + a[n-2] + 2*a[n-1]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Nov 23 2016 *)
%Y For sequences b[n], c[n], d[n] and a[n]+b[n]+c[n]+d[n] see A126941, A126942, A126943, A126944 respectively.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Mar 18 2007
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