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A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.
(Formerly M3865)
14
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 (list; graph; refs; listen; history; text; internal format)
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).

D n Verma, Towards Classifying Finite Point-Set Configurations, preprint, 1997.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..710

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 (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(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

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

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

Sequence in context: A011863 A027085 A099452 * A001753 A202087 A073459

Adjacent sequences:  A006004 A006005 A006006 * A006008 A006009 A006010

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Henry Bottomley, Apr 24 2001

STATUS

approved

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Last modified April 23 02:15 EDT 2019. Contains 322380 sequences. (Running on oeis4.)