A126939 "Model 1" for number of free alkanes on n points.

1, 1, 3, 11, 35, 107, 339, 1073, 3375, 10633, 33525, 105651, 332941, 1049305, 3306957, 10421967, 32845327, 103513709, 326228241, 1028123557, 3240180157, 10211580633, 32182277499, 101423965833, 319642412979, 1007368140211, 3174768208785, 10005431759263

Offset

0,3

Comments

Linear recurrence and empirical g.f confirmed for more terms. - Ray Chandler, Mar 07 2024

References

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).

Links

Ray Chandler, Table of n, a(n) for n = 0..2006

Index entries for linear recurrences with constant coefficients, signature (2, 1, 7, 3, 3, 1).

Formula

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].

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

Maple

M:=35; a:=array(-5..M); b:=array(-5..M); c:=array(-5..M); d:=array(-5..M);
for i from -5 to 0 do a[i]:=0; b[i]:=0; c[i]:=0; d[i]:=0; od: a[0]:=1;
for n from 1 to M do
a[n]:=a[n-1]+b[n-1]+d[n-1];
b[n]:=2*a[n-1]+b[n-1]+b[n-3]+c[n-3]+c[n-4];
c[n]:=2*a[n-1]+b[n-1]+b[n-2]+b[n-3]+2*c[n-3]+c[n-4];
d[n]:=b[n-1]+b[n-2]+c[n-1]+2*c[n-2]+c[n-3]; od:

Mathematica

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 *)

Crossrefs

For sequences b[n], c[n], d[n] and a[n]+b[n]+c[n]+d[n] see A126941, A126942, A126943, A126944 respectively.

Sequence in context: A320683 A027060 A171498 * A370199 A126940 A026125

Adjacent sequences: A126936 A126937 A126938 * A126940 A126941 A126942

Keyword

nonn

Author

N. J. A. Sloane, Mar 18 2007

Status

approved