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%I M3413 #112 Aug 23 2022 09:50:03
%S 1,0,0,1,4,11,25,50,91,154,246,375,550,781,1079,1456,1925,2500,3196,
%T 4029,5016,6175,7525,9086,10879,12926,15250,17875,20826,24129,27811,
%U 31900,36425,41416,46904,52921,59500,66675,74481,82954,92131
%N 4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n)=coeff(charpoly(A,x),x^(n-4)). - _Milan Janjic_, Jan 24 2010
%C From _Ant King_, Sep 14 2011: (Start)
%C Consider the array formed by the polygonal numbers of increasing rank A139600
%C 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... A000217(n)
%C 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ... A000290(n)
%C 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, ... A000326(n)
%C 0, 1, 6, 15, 28, 45, 66, 91, 120, 153, ... A000384(n)
%C 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, ... A000566(n)
%C 0, 1, 8, 21, 40, 65, 96, 133, 176, 225, ... A000567(n)
%C ...
%C Then, for n>=2, a(n) is the diagonal sum of this polygonal grid.
%C (End)
%C Binomial transform of (1, -1, 1, 0, 1, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 26 2015
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
%D Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 102.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006522/b006522.txt">Table of n, a(n) for n = 0..1000</a>
%H J. W. Freeman, <a href="http://www.jstor.org/stable/2689875">The number of regions determined by a convex polygon</a>, Math. Mag., 49 (1976), 23-25.
%H Math Forum, <a href="https://web.archive.org/web/20171019225455/http://mathforum.org:80/library/drmath/view/55262.html">Regions of a circle Cut by Chords to n points</a>.
%H V. Meally, <a href="/A006516/a006516.pdf">Letter to N. J. A. Sloane, May 1975</a>.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonal.html">Polygon Diagonal</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(n,4) + binomial(n-1,2) = A000332(n) + A000217(n-2).
%F a(n) = binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4). - _Zerinvary Lajos_, Jul 23 2006
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=4. - _Harvey P. Dale_, Jul 11 2011
%F G.f.: -((x-1)*x*(x*(4*x-5)+5)+1)/(x-1)^5. - _Harvey P. Dale_, Jul 11 2011
%F a(n) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 24)/24. - _T. D. Noe_, Oct 16 2013
%F For odd n, a(n) = A007678(n). - _R. J. Mathar_, Nov 22 2017
%F a(n) = a(3-n) for all n in Z. - _Michael Somos_, Nov 23 2021
%F Sum_{n>=3} 1/a(n) = 66/25 - (4/5)*sqrt(3/13)*Pi*tanh(sqrt(39)*Pi/2). - _Amiram Eldar_, Aug 23 2022
%e For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).
%p A006522 := n->(1/24)*(n-1)*(n-2)*(n^2-3*n+12):
%p seq(A006522(n), n=0..40);
%p A006522:=-(1-z+z**2)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation; gives sequence except for three leading terms
%t a=2;b=3;s=4;lst={1,0,0,1,s};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,2,6!,1}];lst (* _Vladimir Joseph Stephan Orlovsky_, May 24 2009 *)
%t Table[Binomial[n,4]+Binomial[n-1,2],{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,0,0,1,4},40] (* _Harvey P. Dale_, Jul 11 2011 *)
%t CoefficientList[Series[-(((x - 1) x (x (4 x - 5) + 5) + 1) / (x - 1)^5), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t a[n_] := If[n==0, 1, Sum[PolygonalNumber[n-k+1, k], {k, 0, n-2}]];
%t a /@ Range[0, 40] (* _Jean-François Alcover_, Jan 21 2020 *)
%o (Magma) [Binomial(n, 4)+Binomial(n-1, 2): n in [0..40]]; // _Vincenzo Librandi_, Jun 09 2013
%o (PARI) a(n)=1/24*n^4 - 1/4*n^3 + 23/24*n^2 - 7/4*n + 1 \\ _Charles R Greathouse IV_, Feb 09 2017
%Y Partial sums of A004006.
%Y Cf. A000332, A000217.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
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