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%I #31 Mar 16 2018 16:58:10
%S 1,4,20,60,170,396,868,1716,3235,5720,9752,15912,25236,38760,58200,
%T 85272,122661,173052,240460,328900,444158,592020,780572,1017900,
%U 1315015,1682928,2136304,2689808,3362600,4173840,5148144,6310128
%N Alkane (or paraffin) numbers l(10,n).
%C 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.
%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D Winston C. Yang (paper in preparation).
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).
%F G.f.: (1+6*x^2+x^4)/((1-x)^4*(1-x^2)^4). [ _N. J. A. Sloane_ ]
%F 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.
%F 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]
%p 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
%t 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 *)
%Y Cf. A282011.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)
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