A018211 Alkane (or paraffin) numbers l(10,n).

1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912, 25236, 38760, 58200, 85272, 122661, 173052, 240460, 328900, 444158, 592020, 780572, 1017900, 1315015, 1682928, 2136304, 2689808, 3362600, 4173840, 5148144, 6310128

Offset

0,2

Comments

Equals (1/2) * ((1, 8, 36, 120, 330, 792,...) + (1, 0, 4, 0, 10, 0, 20,...)); where (1, 8, 36,..) = A000580 = C(n,7), and (1, 4, 10,...) = the Tetrahedral numbers.

References

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

Winston C. Yang (paper in preparation).

Links

Table of n, a(n) for n=0..31.

N. J. A. Sloane, Classic Sequences

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).

Formula

G.f.: (1+6*x^2+x^4)/((1-x)^4*(1-x^2)^4). [ N. J. A. Sloane ]

l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.

a(n) = (1/(2*7!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7) + (1/3)*(1/2^5)*(n+2)*(n+4)*(n+6)*(1/2)*(1+(-1)^n) [Yosu Yurramendi Jun 23 2013]

Maple

a:= n-> (Matrix([[1, 0$7, -1, -4, -20, -60]]). Matrix(12, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1][i], 0)))^n)[1, 1]: seq(a(n), n=0..31); # Alois P. Heinz, Jul 31 2008

Mathematica

LinearRecurrence[{4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1}, {1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912}, 32] (* Ray Chandler, Sep 23 2015 *)

Crossrefs

Cf. A282011.

Sequence in context: A196213 A196680 A033488 * A135507 A197404 A355990

Adjacent sequences: A018208 A018209 A018210 * A018212 A018213 A018214

Keyword

nonn

Author

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved