A005999 Number of paraffins.

1, 2, 6, 11, 23, 38, 64, 95, 141, 194, 266, 347, 451, 566, 708, 863, 1049, 1250, 1486, 1739, 2031, 2342, 2696, 3071, 3493, 3938, 4434, 4955, 5531, 6134, 6796, 7487, 8241, 9026, 9878, 10763, 11719, 12710, 13776, 14879, 16061, 17282, 18586, 19931, 21363, 22838, 24404, 26015, 27721, 29474, 31326, 33227, 35231, 37286

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

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

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Crossrefs

Cf. A005997.

Programs

Magma

[1+Floor((n-1)/2)+2*(Binomial(n+1,3)-Binomial(Floor((n+1)/2),3)-Binomial(Ceiling((n+1)/2),3))-(n-1)^2 : n in [1..50]]; // Wesley Ivan Hurt, Sep 16 2014

Maple

A005999:=n->1+floor((n-1)/2)+2*(binomial(n+1,3)-binomial(floor((n+1)/2),3)-binomial(ceil((n+1)/2),3))-(n-1)^2: seq(A005999(n), n=1..40); # Wesley Ivan Hurt, Sep 16 2014

Mathematica

A005997[n_] := 1 + Floor[(n-1)/2] + 2*(Binomial[n+1,3] -Binomial[Floor[(n+1)/2],3] - Binomial[Ceiling[(n+1)/2],3]); A005999[n_] := A005997[n] - (n-1)^2; Array[A005999, 100] (* Enrique Pérez Herrero, Apr 22 2012 *)

PARI

Vec( (x^5+2*x^4+x^3+x^2+1)/(-1+x)^2/(-1+x^2)^2 + O(x^66) ) \\ Joerg Arndt, Sep 16 2014

Formula

G.f.: (x^5+2*x^4+x^3+x^2+1)/((-1+x)^2*(-1+x^2)^2).

a(n) = A005997(n) - (n-1)^2. - Enrique Pérez Herrero, Mar 28 2012