A006009 -id:A006009 - cp's OEIS frontend

A005994 Alkane (or paraffin) numbers l(7,n).

1, 3, 9, 19, 38, 66, 110, 170, 255, 365, 511, 693, 924, 1204, 1548, 1956, 2445, 3015, 3685, 4455, 5346, 6358, 7514, 8814, 10283, 11921, 13755, 15785, 18040, 20520, 23256, 26248, 29529, 33099, 36993, 41211, 45790, 50730, 56070, 61810, 67991

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Equals A000217 (1, 3, 6, 10, 15, ...) convolved with A193356 (1, 0, 3, 0, 5, ...). - Gary W. Adamson, Feb 16 2009
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;
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
a(A254338(n)) = 6 for n > 0. - Reinhard Zumkeller, Feb 27 2015

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
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.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences
Ata Aydin Uslu and Hamdi G. Ozmenekse, Number of bracelets made with 1 blue, 4 red and n black beads.
Ata Aydin Uslu and Hamdi G. Ozmenekse, Number of bracelets made with 1 blue, 2 red and n black beads.
Index entries for linear recurrences with constant coefficients, signature (3, -1, -5, 5, 1, -3, 1).

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

Cf. A000217, A005408, A282011.

--  Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005994 n = a005994_list !! n
a005994_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
                   tail $ inits $ tail a000217_list
-- Reinhard Zumkeller, Feb 27 2015

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

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 *)
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 *)

{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 */

G.f.: (1+x^2)/((1-x)^3*(1-x^2)^2) = (1+x^2)/((1-x)^5*(1+x)^2).
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.
a(-5-n)=a(n). - Michael Somos, Mar 08 2007
Euler transform of length 4 sequence [3, 3, 0, -1]. - Michael Somos, Mar 08 2007
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
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
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
Conjecture: a(n)+a(n+1) = A203286(n+1). - R. J. Mathar, Mar 08 2025

A007009 Number of 3-voter voting schemes with n linearly ranked choices.

Original entry on oeis.org

1, 4, 12, 27, 54, 96, 160, 250, 375, 540, 756, 1029, 1372, 1792, 2304, 2916, 3645, 4500, 5500, 6655, 7986, 9504, 11232, 13182, 15379, 17836, 20580, 23625, 27000, 30720, 34816, 39304, 44217, 49572, 55404, 61731, 68590, 76000, 84000, 92610, 101871, 111804

Offset: 1

Daniel E. Loeb

With a(0) = 0 nontrivial integer solutions of (x + y)^3 = (x - y)^4. If x = a(n) then y = a(n + (-1)^n). - Thomas Scheuerle, Mar 22 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A034828 (first differences).

I:=[1,4,12,27,54,96,160]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)- 5*Self(n-3)+5*Self(n-4)+Self(n-5)-3*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Sep 21 2015

a:= n-> (Matrix([[0$4, 1, 4, 12, 27]]). Matrix(8, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [4, -4, -4, 10, -4, -4, 4, -1][i], 0)))^n)[1, 1]:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 13 2008

LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 4, 12, 27, 54, 96, 160}, 50] (* Vincenzo Librandi, Sep 21 2015 *)

Vec(x*(1-x^3)/((1-x)^4*(1-x^2)^2) + O(x^100)) \\ Colin Barker, Jan 07 2016

G.f.: x*(1-x^3)/((1-x)^4*(1-x^2)^2) = x*(1+x+x^2)/((1-x)^5*(1+x)^2).
a(n) = (1/2)*Sum_{k=1..n+1} k*floor(k/2)*ceiling(k/2). - Vladeta Jovovic, Apr 29 2006
a(n) = A006009(n)/4.
a(n) = A007590(n+2)*A007590(n+1)/8. - Richard R. Forberg, Dec 03 2013
For n > 1, a(n) = A000332(n+3) - A002624(n-2). - Antal Pinter, Sep 20 2015
a(n) = (n^4 + 6*n^3 + 12*n^2 + 8*n)/32 for n even; a(n) = (n^4 + 6*n^3 + 12*n^2 + 10*n + 3)/32 for n odd. - Colin Barker, Jan 07 2016

More terms from James Sellers, Sep 08 2000

cp's OEIS Frontend

A005994 Alkane (or paraffin) numbers l(7,n).

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