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A005995
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Alkane (or paraffin) numbers l(8,n).
(Formerly M2916)
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5
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1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, 10197, 13167, 16852, 21252, 26598, 32890, 40404, 49140, 59423, 71253, 85008, 100688, 118728, 139128, 162384, 188496, 218025
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from M. F. Hasler, May 01 2009: (Start)
Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.
A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End)
Equals (1/2)*((1, 6, 21, 56, 126, 252,...) + (1, 0, 3, 0, 6, 0, 10,...)).
Equals row sums of triangle A160770.
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REFERENCES
| S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Smith, Polynomial invariants of finite groups, Bull. Am. Math. Soc. 34 (1997), 211-250.
Winston C. Yang (paper in preparation).
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LINKS
| N. J. A. Sloane, Classic Sequences
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FORMULA
| G.f.: (1+3*x^2)/((1-x)^3*(1-x^2)^3).
a(2n-1) = n(n+1)(n+2)(2n+1)(2n+3)/30, a(2n) = (n+1)(n+2)(n+3)(4n^2+6n+5)/ 30. [From M. F. Hasler, May 01 2009]
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MAPLE
| a:= n-> (Matrix([[1, 0$6, -3, -9]]). Matrix(9, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=0..32); # Alois P. Heinz, Jul 31 2008
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PROG
| (PARI) a(k)= if(k%2, (k+1)*(k+3)*(k+5), (k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 [From M. F. Hasler, May 01 2009]
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CROSSREFS
| Cf. A160770.
Sequence in context: A066643 A140065 A115549 * A034503 A026557 A124052
Adjacent sequences: A005992 A005993 A005994 * A005996 A005997 A005998
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Winston C. Yang (yang(AT)math.wisc.edu)
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