A005997 Number of paraffins. (Formerly M2832)

1, 3, 10, 20, 39, 63, 100, 144, 205, 275, 366, 468, 595, 735, 904, 1088, 1305, 1539, 1810, 2100, 2431, 2783, 3180, 3600, 4069, 4563, 5110, 5684, 6315, 6975, 7696, 8448, 9265, 10115, 11034, 11988, 13015, 14079, 15220, 16400, 17661, 18963, 20350, 21780, 23299

Offset

1,2

References

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).

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. (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

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

a(n) = A005999(n)+(n-1)^2. - Enrique Pérez Herrero, Mar 27 2012

a(n) = 1 + floor((n-1)/2) + 2*(C(n+1,3)-C(floor((n+1)/2),3)-C(ceiling((n+1)/2),3). - Enrique Pérez Herrero, Apr 22 2012

a(n) = (n+1)(2n^2-(-1)^n+1)/8. - Bruno Berselli, Apr 22 2012

a(n) = A004526(n) + 2*A111384(n). - Enrique Pérez Herrero, Apr 25 2012

E.g.f.: (x*(3 + 4*x + x^2)*cosh(x) + (1 + 2*x + 4*x^2 + x^3)*sinh(x))/4. - Stefano Spezia, Dec 13 2021

Maple

a:= n-> (Matrix([[0, 0, -1, -5, -12, -26]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=1..50); # Alois P. Heinz, Jul 31 2008

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]); Array[A005997, 37] (* Enrique Pérez Herrero, Apr 22 2012 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 3, 10, 20, 39, 63}, 37] (* Bruno Berselli, Apr 22 2012 *)

Crossrefs

Cf. A005999.

Sequence in context: A272764 A293765 A295953 * A213850 A336529 A081205

Adjacent sequences: A005994 A005995 A005996 * A005998 A005999 A006000

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved