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, 369741, 411958, 457691, 507130

Offset: 1

Author

Albert Rich (Albert_Rich(AT)msn.com)

Keywords

• nonn
• easy

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

Examples

			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
		

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, and 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• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission].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).

Crossrefs

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

Programs

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

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 *)

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

Sage

[n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # G. C. Greubel, Sep 02 2019

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

From Elmo R. Oliveira, Aug 14 2025: (Start)

E.g.f.: exp(x)*x*(2 + x)*(12 + 30*x + 5*x^2)/24.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)