A005999 Number of paraffins. (Formerly M1577)

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

1,2

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).

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A005997.

Sequence in context: A083322 A073939 A085264 * A102429 A080012 A103143

Adjacent sequences: A005996 A005997 A005998 * A006000 A006001 A006002

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved