A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

0, 1, 5, 16, 40, 85, 161, 280, 456, 705, 1045, 1496, 2080, 2821, 3745, 4880, 6256, 7905, 9861, 12160, 14840, 17941, 21505, 25576, 30200, 35425, 41301, 47880, 55216, 63365, 72385, 82336, 93280, 105281, 118405, 132720, 148296, 165205, 183521

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• easy
• nonn
• nice

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• D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished.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).

Crossrefs

Cf. A003215, A000537, A000578, A005898, A027602. - Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Cf. A000217, A061315, A061316, A061318.

Programs

Magma

[n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

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 *)

PARI

a(n)=n*(n+1)*(n^2+n+4)/12 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: (1+x^2)/(1-x)^5.

a(n) = 2*binomial(n + 2, 4) + binomial(n + 1, 2).

a(n) = A061316(n)/3 = A061315(n, 3) = sqrt(A061318(n)-A061316(n)).

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

Extensions

More terms from Henry Bottomley, Apr 24 2001