A006322 4-dimensional analog of centered polygonal numbers.

1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, 5083, 6916, 9205, 12020, 15436, 19533, 24396, 30115, 36785, 44506, 53383, 63526, 75050, 88075, 102726, 119133, 137431, 157760, 180265, 205096, 232408, 262361, 295120, 330855

Offset

1,2

Comments

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005

Partial sums give A006414. - L. Edson Jeffery, Dec 13 2011

Also the number of (w,x,y,z) with all terms in {1,...,n} and w<=x>=y<=z, see A211795. - Clark Kimberling, May 19 2012

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/4).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 2 Table 2 p. 9.

Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]. See p. 31.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 5*C(n + 2, 4) + C(n + 1, 2) = (C(5*n+4, 4) - 1)/5^3 = n*(n+1)*(5*n^2 + 5*n + 2)/24.

a(n) = (((n+1)^5-n^5) - ((n+1)^3-n^3))/24. - Xavier Acloque, Jan 14 2003, corrected by Eric Rowland, Aug 15 2017

Partial sums of A004068. - Xavier Acloque, Jan 15 2003

G.f.: x*(1+3*x+x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

a(n) = Sum_{i=1..n} Sum_{j=1..n} i * min(i,j). - Enrique Pérez Herrero, Jan 30 2013

a(n) = A000537(n) - A000332(n+2). - J. M. Bergot, Jun 03 2017

Sum_{n>=1} 1/a(n) = 42 - 4*sqrt(15)*Pi*tanh(sqrt(3/5)*Pi/2). - Amiram Eldar, May 28 2022

Example

An illustration for a(5)=190: 5*(1+2+3+4+5)+4*(2+3+4+5)+3*(3+4+5)+2*(4+5)+1*(5) gives 75+56+36+18+5=190. - J. M. Bergot, Feb 13 2018

Maple

a:=n->5*binomial(n+2, 4) + binomial(n+1, 2): seq(a(n), n=1..40); # Muniru A Asiru, Feb 13 2018

Mathematica

Table[5*Binomial[n+2, 4] + Binomial[n+1, 2], {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[(1+3x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 31, 85, 190}, 40] (* Harvey P. Dale, Sep 27 2016 *)

Prog

(PARI) a(n)=n*(5*n^3+10*n^2+7*n+2)/24 \\ Charles R Greathouse IV, Dec 13 2011, corrected by Altug Alkan, Aug 15 2017
(GAP) List([1..40], n->5*Binomial(n+2, 4) + Binomial(n+1, 2)); # Muniru A Asiru, Feb 13 2018
(Magma) [n*(n+1)*(5*n^2 +5*n +2)/24: n in [1..40]]; // G. C. Greubel, Sep 02 2019
(Sage) [n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # G. C. Greubel, Sep 02 2019

Crossrefs

Cf. A000217, A000330, A006414, A050446, A050447.

Sequence in context: A303522 A299261 A005338 * A319906 A212064 A213764

Adjacent sequences: A006319 A006320 A006321 * A006323 A006324 A006325

Keyword

nonn,easy

Author

Albert Rich (Albert_Rich(AT)msn.com)

Status

approved