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A006322 4-dimensional analog of centered polygonal numbers.

%I A006322 #98 Aug 14 2025 15:15:37
%S A006322 1,8,31,85,190,371,658,1086,1695,2530,3641,5083,6916,9205,12020,15436,
%T A006322 19533,24396,30115,36785,44506,53383,63526,75050,88075,102726,119133,
%U A006322 137431,157760,180265,205096,232408,262361,295120,330855,369741,411958,457691,507130
%N A006322 4-dimensional analog of centered polygonal numbers.
%C A006322 Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Nov 18 2005
%C A006322 Partial sums give A006414. - _L. Edson Jeffery_, Dec 13 2011
%C A006322 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
%D A006322 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).
%H A006322 Vincenzo Librandi, <a href="/A006322/b006322.txt">Table of n, a(n) for n = 1..1000</a>
%H A006322 Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, <a href="https://arxiv.org/abs/1811.05707">Plateau Polycubes and Lateral Area</a>, arXiv:1811.05707 [math.CO], 2018. See Column 2 Table 2 p. 9.
%H A006322 Manfred Goebel, <a href="http://dx.doi.org/10.1007/s002000050118">Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials</a>, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%H A006322 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]. See p. 31.
%H A006322 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A006322 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.
%F A006322 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
%F A006322 Partial sums of A004068. - Xavier Acloque, Jan 15 2003
%F A006322 G.f.: x*(1+3*x+x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F A006322 a(n) = Sum_{i=1..n} Sum_{j=1..n} i * min(i,j). - _Enrique Pérez Herrero_, Jan 30 2013
%F A006322 a(n) = A000537(n) - A000332(n+2). - _J. M. Bergot_, Jun 03 2017
%F A006322 Sum_{n>=1} 1/a(n) = 42 - 4*sqrt(15)*Pi*tanh(sqrt(3/5)*Pi/2). - _Amiram Eldar_, May 28 2022
%F A006322 From _Elmo R. Oliveira_, Aug 14 2025: (Start)
%F A006322 E.g.f.: exp(x)*x*(2 + x)*(12 + 30*x + 5*x^2)/24.
%F A006322 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)
%e A006322 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
%p A006322 a:=n->5*binomial(n+2,4) + binomial(n+1,2): seq(a(n), n=1..40); # _Muniru A Asiru_, Feb 13 2018
%t A006322 Table[5*Binomial[n+2, 4] + Binomial[n+1, 2], {n, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t A006322 CoefficientList[Series[(1+3x+x^2)/(1-x)^5, {x,0,40}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t A006322 LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190},40] (* _Harvey P. Dale_, Sep 27 2016 *)
%o A006322 (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
%o A006322 (GAP) List([1..40], n->5*Binomial(n+2,4) + Binomial(n+1,2)); # _Muniru A Asiru_, Feb 13 2018
%o A006322 (Magma) [n*(n+1)*(5*n^2 +5*n +2)/24: n in [1..40]]; // _G. C. Greubel_, Sep 02 2019
%o A006322 (Sage) [n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # _G. C. Greubel_, Sep 02 2019
%Y A006322 Cf. A000217, A000330, A006414, A050446, A050447.
%K A006322 nonn,easy
%O A006322 1,2
%A A006322 Albert Rich (Albert_Rich(AT)msn.com)