A005994 - OEIS

%I M2774 #78 Mar 30 2018 19:50:30

%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 A000217 (1, 3, 6, 10, 15, ...) convolved with A193356 (1, 0, 3, 0, 5, ...). - _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)]. - _Ata Aydin Uslu_ and Hamdi G. Ozmenekse, Jan 11 2012

%C a(A254338(n)) = 6 for n > 0. - _Reinhard Zumkeller_, Feb 27 2015

%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 Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-Szegö 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 <a href="/index/Rec#order_07">Index entries for 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 Aydin Uslu_ and Hamdi G. Ozmenekse, 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 Aydin Uslu_ and Hamdi G. Ozmenekse, Jan 12 2012].

%F G.f.: (1+x^2)/((1-x)^3*(1-x^2)^2) = (1+x^2)/((1-x)^5*(1+x)^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(n) = 3a(n-1) - a(n-2) - 5a(n-3) + 5a(n-4) + a(n-5) - 3a(n-6) + a(n-7), with a(0)=1, a(1)=3, a(2)=9, a(4)=19, a(5)=38, a(6)=66, a(7)=110. - _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)) - binomial(n+4,4). - _Enrique Pérez Herrero_, May 11 2012

%F a(n) = (1/48)*(n+1)*(n+3)*((n+2)*(n+4)+3)+1/32*(2*n+5)*(1+(-1)^n). - _Yosu Yurramendi_, Jun 20 2013

%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] (* _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}]]] (* _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 */

%o (Haskell)   Following _Gary W. Adamson_.

%o import Data.List (inits, intersperse)

%o a005994 n = a005994_list !! n

%o a005994_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $

%o                    tail $ inits $ tail a000217_list

%o -- _Reinhard Zumkeller_, Feb 27 2015

%Y Cf. A006009, A005997, A005993 (first differences).

%Y Cf. A000217, A005408, A282011.

%K nonn,easy,nice

%O 0,2

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