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%I #55 May 08 2020 09:36:42
%S 1,4,16,44,110,236,472,868,1519,2520,4032,6216,9324,13608,19440,27192,
%T 37389,50556,67408,88660,115258,148148,188552,237692,297115,368368,
%U 453376,554064,672792,811920,974304,1162800,1380825,1631796
%N Alkane (or paraffin) numbers l(9,n).
%C From _M. F. Hasler_, May 02 2009: (Start)
%C Also, 6th column of A159916, i.e., number of 6-element subsets of {1,...,n+6} whose elements add up to an odd integer.
%C Third differences are A002412([n/2]). (End)
%C F(1,6,n) is the number of bracelets with 1 blue, 6 identical red and n identical black beads. If F(1,6,1) = 4 and F(1,6,2) = 16 taken as a base, F(1,6,n) = n(n+1)(n+2)(n+3)(n+4)/120 + F(1,4,n) + F(1,6,n-2). F(1,4,n) is the number of bracelets with 1 blue, 4 identical red and n identical black beads. If F(1,4,1) = 3 and F(1,4,2) = 9 taken as a base; F(1,4,n) = n(n+1)(n+2)/6 + F(1,2,n) + F(1,4,n-2). F(1,2,n) is the number of bracelets with 1 blue, 2 identical red and n identical black beads. If F(1,2,1) = 2 and F(1,2,2) = 4 taken as a base F(1,2,n) = n + 1 + F(1,2,n-2). - _Ata Aydin Uslu_ and Hamdi G. Ozmenekse, Mar 16 2012
%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 Ata A. Uslu and Hamdi G. Ozmenekse, <a href="http://upload.wikimedia.org/wikipedia/commons/3/34/F%281%2C6%2Cn%29.pdf">F(1,6,n)</a>
%H Ata A. Uslu and Hamdi G. Ozmenekse, <a href="http://upload.wikimedia.org/wikipedia/commons/9/90/Bracelet_Problem_%28Bileklik_problemi%29.pdf">F(1,4,n)</a>
%H Ata A. Uslu and Hamdi G. Ozmenekse, <a href="http://upload.wikimedia.org/wikipedia/commons/6/64/Bileklik_Problemi_%28Bracelet_Problem%29.pdf">F(1,2,n)</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4, -3, -8, 14, 0, -14, 8, 3, -4, 1).
%F G.f.: (1+3*x^2)/(1-x)^4/(1-x^2)^3. - _N. J. A. Sloane_
%F l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
%F a(2n) = (n+1)(n+2)(n+3)^2(4n^2+6n+5)/90, a(2n-1) = n(n+1)(n+2)(n+3)(4n^2+6n+5)/90. - _M. F. Hasler_, May 02 2009
%F a(n) = (1/(2*6!))*(n+2)*(n+4)*(n+6)*((n+1)*(n+3)*(n+5) + 1*3*5) - (1/2)*(1/2^4)*(n^2+7*n+11)*(1/2)*(1-(-1)^n). - _Yosu Yurramendi_, Jun 23 2013
%F a(n) = A060099(n)+3*A060099(n-2). - _R. J. Mathar_, May 08 2020
%p a:=n-> (Matrix([[1,0$7,3,12]]). Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [4, -3, -8, 14, 0, -14, 8, 3, -4, 1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..33); # _Alois P. Heinz_, Jul 31 2008
%t CoefficientList[(1+3*x^2)/((1-x)^7*(1+x)^3) + O[x]^34, x] (* _Jean-François Alcover_, Jun 08 2015 *)
%t LinearRecurrence[{4, -3, -8, 14, 0, -14, 8, 3, -4, 1},{1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520},34] (* _Ray Chandler_, Sep 23 2015 *)
%o (PARI) A018210(n)=(n+2)*(n+4)*(n+6)^2*(n^2+3*n+5)/1440-if(n%2,(n^2+7*n+11)/32) \\ _M. F. Hasler_, May 02 2009
%Y Cf. A005995 (first differences).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)
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