A005998 Number of paraffins.

1, 2, 7, 14, 29, 48, 79, 116, 169, 230, 311, 402, 517, 644, 799, 968, 1169, 1386, 1639, 1910, 2221, 2552, 2927, 3324, 3769, 4238, 4759, 5306, 5909, 6540, 7231, 7952, 8737, 9554, 10439, 11358, 12349, 13376, 14479, 15620, 16841, 18102, 19447, 20834, 22309

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
• 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).

Programs

Magma

[1/8*(2*n^3-2*n^2+5*n-(-1)^n*(n+1)+1): n in [1..50]]; // Vincenzo Librandi, Mar 15 2014

Maple

a:= n-> (Matrix([[0, -1, -4, -11, -22, -41]]). 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..38); # Alois P. Heinz, Jul 31 2008

Mathematica

a[n_] := 1/8*(2*n^3-2*n^2+5*n-(-1)^n*(n+1)+1); Array[a, 40] (* Jean-François Alcover, Mar 13 2014 *)
CoefficientList[Series[(x^4 + 2 x^3 + 2 x^2 + 1)/(-1 + x)^2/(-1 + x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,2,7,14,29,48},50] (* Harvey P. Dale, Oct 13 2024 *)

Formula

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

Extensions

More terms from Vincenzo Librandi, Mar 15 2014