A114289 Number of combinatorial types of n-dimensional polytopes with n+3 vertices.
0, 1, 7, 31, 116, 379, 1133, 3210, 8803, 23701, 63239, 168287, 447905, 1194814, 3196180, 8576505, 23081668, 62292381, 168536249, 457035453, 1241954405, 3381289332, 9221603416, 25189382006, 68906572413, 188750887991
Offset: 1
Keywords
References
- B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
Links
- Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001. See Table 7.5.
- Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
- Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
- E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.
- Aleksandr Maksimenko, 2-neighborly 0/1-polytopes of dimension 7, arXiv:1904.03638 [math.CO], 2019.
Programs
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Maple
N:=60: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-ln(1-2*x)+ln(1-x): K:=-1/2*x*(x-8*x^3-1+5*x^2-7*x^4+2*x^6+5*x^8-9*x^7+19*x^5-14*x^9+x^10+19*x^11-5*x^12+4*x^14-8*x^13)/(1-x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^2: series(1/(x^3-x^4)*(1/4*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+1/2*sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);
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Mathematica
terms = 26; G[x_] = -Log[1 - 2(x^3/(1 - 2x)^2)]; H[x_] = -Log[1 - 2x] + Log[1 - x]; K[x_] = -1/2 x (x - 8x^3 - 1 + 5x^2 - 7x^4 + 2x^6 + 5x^8 - 9x^7 + 19x^5 - 14x^9 + x^10 + 19x^11 - 5x^12 + 4x^14 - 8x^13)/(1-x)^5/(2x^6 - 4x^4 + 4x^2 - 1)/(x+1)^2; 1/(x^3 - x^4) (1/4 Sum[EulerPhi[2r + 1]/(2r + 1) G[x^(2r + 1)], {r, 0, terms+2}] + 1/2 Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+2}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* Jean-François Alcover, Dec 14 2018 *)