A114291 -id:A114291 - cp's OEIS frontend

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

Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005

nonn

B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.

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.

Cf. A000943, A114290, A114291.

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);

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 *)

A114290 Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.

Original entry on oeis.org

0, 1, 7, 38, 170, 617, 1979, 5859, 16571, 45516, 123159, 330736, 885780, 2372305, 6362965, 17102719, 46078541, 124440388, 336829857, 913658780, 2483217288, 6761405513, 18441239903, 50375429081, 137807555515, 377492301876

Offset: 1

Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005

nonn

B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.

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.

Cf. A000943, A114289, A114291.

N:=30: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-log(1-2*x)+ln(1-x): K:=-(x^10+3*x^9-3*x^8-7*x^7+4*x^6+4*x^5+4*x^4+3*x^3-2*x^2+1)*x/(1-x)^5/(x+1)^3: series(1/(x^3-x^4)*(1/2*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);

terms = 26;
G[x_] = -Log[1 - 2 (x^3/(1 - 2 x)^2)];
H[x_] = -Log[1 - 2 x] + Log[1 - x];
K[x_] = -(x^10 + 3 x^9 - 3 x^8 - 7 x^7 + 4 x^6 + 4 x^5 + 4 x^4 + 3 x^3 - 2 x^2 + 1) x/(1 - x)^5/(x + 1)^3;
1/(x^3 - x^4) (1/2 Sum[EulerPhi[2 r + 1]/(2 r + 1) G[x^(2 r + 1)], {r, 0, terms+3}] + Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+3}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* Jean-François Alcover, Dec 14 2018 *)

cp's OEIS Frontend

A114289 Number of combinatorial types of n-dimensional polytopes with n+3 vertices.

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