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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved