OFFSET
1,2
COMMENTS
Kekulé numbers for certain benzenoids. -Emeric Deutsch,Nov 18 2005
Partial sums giveA006414.-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, seeA211795.-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R. P. Stanley,Examples of Magic Labelings,Unpublished Notes, 1973 [Cached copy, with permission].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 byEric Rowland,Aug 15 2017
Partial sums ofA004068.- 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
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 byAltug 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
KEYWORD
nonn,easy
AUTHOR
Albert Rich (Albert_Rich(AT)msn.com)
STATUS
approved