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 A005995 Alkane (or paraffin) numbers l(8,n). (Formerly M2916) 8

%I M2916

%S 1,3,12,28,66,126,236,396,651,1001,1512,2184,3108,4284,5832,7752,

%T 10197,13167,16852,21252,26598,32890,40404,49140,59423,71253,85008,

%U 100688,118728,139128,162384,188496,218025,250971,287964,329004,374794,425334,481404,543004

%N Alkane (or paraffin) numbers l(8,n).

%C From _M. F. Hasler_, May 01 2009: (Start)

%C Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.

%C 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)

%C Equals (1/2)*((1, 6, 21, 56, 126, 252, ...) + (1, 0, 3, 0, 6, 0, 10, ...)), see A000389 and A000217.

%C Equals row sums of triangle A160770.

%C F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x<a+1, [|x|]=a. - Ata A. 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Winston C. Yang (paper in preparation).

%H Alois P. Heinz, <a href="/A005995/b005995.txt">Table of n, a(n) for n = 0..1000</a>

%H Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-SzegĂ¶ polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.

%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 N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>

%H L. Smith, <a href="http://dx.doi.org/10.1090/S0273-0979-97-00724-6">Polynomial invariants of finite groups. A survey of recent developments</a>. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250.

%H Ata A. Uslu and Hamdi G. Ozmenekse, <a href="http://upload.wikimedia.org/wikipedia/commons/1/1c/F%281%2C3%2Cn%29.pdf">F(1,3,n)</a>

%H Ata A. Uslu and Hamdi G. Ozmenekse, <a href="http://upload.wikimedia.org/wikipedia/commons/2/26/F%281%2C5%2Cn%29.pdf">F(1,5,n)</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -8, 6, 6, -8, 0, 3, -1).

%F G.f.: (1+3*x^2)/((1-x)^3*(1-x^2)^3).

%F 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. [_M. F. Hasler_, May 01 2009]

%F a(n) = (1/240)*(n+1)*(n+2)*(n+3)*(n+4)*(n+5) + (1/16)*(n+2)*(n+4)*(1/2)*(1+(-1)^n). [_Yosu Yurramendi_, Jun 20 2013]

%F a(n) = A038163(n)+3*A038163(n-2). - _R. J. Mathar_, Mar 29 2018

%p 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..40); # _Alois P. Heinz_, Jul 31 2008

%t a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* _Jean-FranĂ§ois Alcover_, Mar 17 2014, after _M. F. Hasler_ *)

%t LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1},{1, 3, 12, 28, 66, 126, 236, 396, 651},40] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) a(k)= if(k%2,(k+1)*(k+3)*(k+5),(k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ _M. F. Hasler_, May 01 2009

%Y Cf. A160770, A053132 (bisection), A271870 (bisection), A018210 (partial sums).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)

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Last modified November 14 17:12 EST 2018. Contains 317210 sequences. (Running on oeis4.)