OFFSET
0,3
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
Vincenzo Librandi, Table of n, a(n) for n = 0..710
Per Alexandersson, Sam Hopkins, and Gjergji Zaimi, Restricted Birkhoff polytopes and Ehrhart period collapse, arXiv:2206.02276 [math.CO], 2022.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - N. J. A. Sloane, Oct 04 2021]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (1+x^2)/(1-x)^5.
a(n) = 2*binomial(n + 2, 4) + binomial(n + 1, 2).
a(0)=0, a(1)=1, a(2)=5, a(3)=16, a(4)=40, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 30 2011
For n>0, a(n) = (A000217(n-1)^2 + A000217(n)^2 + A000217(n+1)^2 - 1)/9. - Richard R. Forberg, Dec 25 2013
Sum_{n>=1} 1/a(n) = 15/4 - tanh(sqrt(15)*Pi/2)*Pi*sqrt(3/5). - Amiram Eldar, Aug 23 2022
E.g.f.: exp(x)*(12 + 48*x + 42*x^2 + 12*x^3 + x^4)/12. - Stefano Spezia, Aug 31 2023
MATHEMATICA
f[n_]:=n^3; lst={}; s=0; Do[s+=(f[n]+f[n+1]+f[n+2]); AppendTo[lst, s/9], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Table[2Binomial[n+2, 4]+Binomial[n+1, 2], {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 5, 16, 40}, 40] (* Harvey P. Dale, Sep 30 2011 *)
PROG
(Magma) [n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n)=n*(n+1)*(n^2+n+4)/12 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
More terms from Henry Bottomley, Apr 24 2001
STATUS
approved