A018210 Alkane (or paraffin) numbers l(9,n).

1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520, 4032, 6216, 9324, 13608, 19440, 27192, 37389, 50556, 67408, 88660, 115258, 148148, 188552, 237692, 297115, 368368, 453376, 554064, 672792, 811920, 974304, 1162800, 1380825, 1631796

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

From M. F. Hasler, May 02 2009: (Start)

Also, 6th column of A159916, i.e., number of 6-element subsets of {1,...,n+6} whose elements add up to an odd integer.

Third differences are A002412([n/2]). (End)

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

References

• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
• Winston C. Yang (paper in preparation).

Links

• N. J. A. Sloane, Classic Sequences
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
• Ata A. Uslu and Hamdi G. Ozmenekse, F(1,6,n)
• Ata A. Uslu and Hamdi G. Ozmenekse, F(1,4,n)
• Ata A. Uslu and Hamdi G. Ozmenekse, F(1,2,n)
• Index entries for linear recurrences with constant coefficients, signature (4, -3, -8, 14, 0, -14, 8, 3, -4, 1).

Crossrefs

Cf. A005995 (first differences).

Programs

Maple

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

Mathematica

CoefficientList[(1+3*x^2)/((1-x)^7*(1+x)^3) + O[x]^34, x] (* Jean-François Alcover, Jun 08 2015 *)
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 *)

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

Formula

G.f.: (1+3*x^2)/(1-x)^4/(1-x^2)^3. - N. J. A. Sloane

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(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

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

a(n) = A060099(n)+3*A060099(n-2). - R. J. Mathar, May 08 2020