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A018211 Alkane (or paraffin) numbers l(10,n). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A196213 A196680 A033488 * A135507 A197404 A169637

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

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Last modified December 4 17:40 EST 2016. Contains 278755 sequences.