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%I M2774
%S 1,3,9,19,38,66,110,170,255,365,511,693,924,1204,1548,1956,2445,3015,
%T 3685,4455,5346,6358,7514,8814,10283,11921,13755,15785,18040,20520,
%U 23256,26248,29529,33099,36993,41211,45790,50730,56070,61810,67991
%N Alkane (or paraffin) numbers l(7,n).
%C Equals (1, 3, 6, 10, 15,...) convolved with (1, 0, 3, 0, 5,...). [From _Gary W. Adamson_, Feb 16 2009]
%C F(1,4,n) is the number of bracelets with 1 blue, 4 red and n black beads. If F(1,4,1)=3 and F(1,4,2)=9 taken as a base;
%C 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 red and n 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)] [From Ata A. Uslu and Hamdi G. Ozmenekse, Jan 11 2012].
%D S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
%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).
%H T. D. Noe, <a href="/A005994/b005994.txt">Table of n, a(n) for n=0..1000</a>
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>
%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%H "http://commons.wikimedia.org/wiki/File:Bracelet_Problem_(Bileklik_problemi).pdf" number of bracelets made with 1 blue, 4 red and n black beads [From Ata A. Uslu and Hamdi G. Ozmenekse (atauslu(AT)hotmail.com.tr), Jan 11 2012].
%H "http://commons.wikimedia.org/wiki/File:Bileklik_Problemi_(Bracelet_Problem).pdf" number of bracelets made with 1 blue, 2 red and n black beads [From Ata A. Uslu and Hamdi G. Ozmenekse (atauslu(AT)hotmail.com.tr), Jan 12 2012].
%F G.f.: (1+x^2)/((1-x)^3*(1-x^2)^2).
%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(-5-n)=a(n) . - Michael Somos, Mar 08 2007
%F Euler transform of length 4 sequence [ 3, 3, 0, -1]. - Michael Somos, Mar 08 2007
%F a(0)=1, a(1)=3, a(2)=9, a(4)=19, a(5)=38, a(6)=66, a(7)=110,
%F a(n)=3a(n-1)-a(n-2)-5a(n-3)+5a(n-4)+a(n-5)-3a(n-6)+a(n-7) [From Harvey P. Dale, May 02 2011]
%F a(n) = A006009(n)/2 - A000332(n+4) = (1/2)*Sum(i=1..n+1, (i+1)*floor((i+1)^2/2)) -C(n+4,4) . - _Enrique Pérez Herrero_, May 11 2012
%p a:= n -> (Matrix([[1, 0$4, 1, 3]]). Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [3, -1, -5, 5, 1, -3, 1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..40); # Alois P. Heinz, Jul 31 2008
%t LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,3,9,19,38,66,110},50] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^3(1-x^2)^2),{x,0,50}],x] (* From Harvey P. Dale, May 02 2011 *)
%t nn=45;With[{a=Accumulate[Range[nn]],b=Riffle[Range[1,nn,2],0]}, Flatten[ Table[ListConvolve[Take[a,n],Take[b,n]],{n,nn}]]] (* From Harvey P. Dale, Nov 11 2011 *)
%o (PARI) {a(n)=if(n<-4, n=-5-n); polcoeff( (1+x^2)/((1-x)^3*(1-x^2)^2)+x*O(x^n), n)} /* Michael Somos, Mar 08 2007 */
%Y Cf. A006009, A005997.
%K nonn,easy,nice,changed
%O 0,2
%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)
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