A006009 Number of paraffins. (Formerly M3513)

4, 16, 48, 108, 216, 384, 640, 1000, 1500, 2160, 3024, 4116, 5488, 7168, 9216, 11664, 14580, 18000, 22000, 26620, 31944, 38016, 44928, 52728, 61516, 71344, 82320, 94500, 108000, 122880, 139264, 157216, 176868, 198288, 221616, 246924, 274360, 304000, 336000

Offset

1,1

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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 (3,-1,-5,5,1,-3,1).

Formula

a(n) = 2*(A005994(n) + binomial(n, 4)).

G.f.: 4*x*(1-x^3) / ((1-x)^4*(1-x^2)^2). - Alois P. Heinz, Aug 13 2008

a(n) = Sum_{i=1..n} i*floor(i^2/2). - Enrique Pérez Herrero, Mar 10 2012

From Enrique Navarrete, Feb 25 2026: (Start)

a(n) = (1/8)*n*(n+2)^3, n even; a(n) = (1/8)*(n+1)^3*(n+3), n odd.

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).

E.g.f.: (1/16)*((2*x^4 + 24*x^3 + 74*x^2 + 56*x + 3)*exp(x) + (2*x - 3)*exp(-x)). (End)

Maple

a:= n-> (Matrix([[0$4, 4, 16, 48, 108]]). Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -4, -4, 10, -4, -4, 4, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=1..40); # Alois P. Heinz, Aug 13 2008

Mathematica

a[n_] := 1/16*(2*n^4+12*n^3+24*n^2+2*(9-(-1)^n)*n-3*(-1)^n+3); Array[a, 40] (* Jean-François Alcover, Mar 17 2014 *)

Crossrefs

4*A007009.

Cf. A005994, A005997.

Sequence in context: A383580 A050616 A297160 * A261977 A100625 A203248

Adjacent sequences: A006006 A006007 A006008 * A006010 A006011 A006012

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved