A006007 - OEIS

%I M3865 #59 Aug 31 2023 15:54:45

%S 0,1,5,16,40,85,161,280,456,705,1045,1496,2080,2821,3745,4880,6256,

%T 7905,9861,12160,14840,17941,21505,25576,30200,35425,41301,47880,

%U 55216,63365,72385,82336,93280,105281,118405,132720,148296,165205,183521

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

%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006007/b006007.txt">Table of n, a(n) for n = 0..710</a>

%H Per Alexandersson, Sam Hopkins, and Gjergji Zaimi, <a href="https://arxiv.org/abs/2206.02276">Restricted Birkhoff polytopes and Ehrhart period collapse</a>, arXiv:2206.02276 [math.CO], 2022.

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

%H D.-N. Verma, <a href="/A012249/a012249.pdf">Towards Classifying Finite Point-Set Configurations</a>, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

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

%F 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

%F For n>0, a(n) = (A000217(n-1)^2 + A000217(n)^2 + A000217(n+1)^2 - 1)/9. - _Richard R. Forberg_, Dec 25 2013

%F Sum_{n>=1} 1/a(n) = 15/4 - tanh(sqrt(15)*Pi/2)*Pi*sqrt(3/5). - _Amiram Eldar_, Aug 23 2022

%F E.g.f.: exp(x)*(12 + 48*x + 42*x^2 + 12*x^3 + x^4)/12. - _Stefano Spezia_, Aug 31 2023

%t 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 *)

%t 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 *)

%o (Magma) [n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011

%o (PARI) a(n)=n*(n+1)*(n^2+n+4)/12 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A003215, A000537, A000578, A005898, A027602. - _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009

%Y Cf. A000217, A061315, A061316, A061318.

%K easy,nonn,nice

%O 0,3

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Henry Bottomley_, Apr 24 2001