This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A140800 #24 Feb 16 2025 08:33:08
%S A140800 1,2,0,50,773,48,83,150,281,540,1055,2082,4133,8232,16427,32814,65585,
%T A140800 131124,262199,524346,1048637,2097216,4194371,8388678,16777289,
%U A140800 33554508,67108943,134217810,268435541,536871000,1073741915,2147483742
%N A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.
%C A140800 _Andrew Weimholt_ suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.
%D A140800 H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
%D A140800 Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
%D A140800 P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.
%H A140800 Georg Fischer, <a href="/A140800/b140800.txt">Table of n, a(n) for n = 0..1000</a>
%H A140800 Gil Kalai, <a href="http://gilkalai.wordpress.com/2008/05/07/five-open-problems-regarding-convex-polytopes/">Five Open Problems Regarding Convex Polytopes</a>.
%H A140800 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Polytope.html">Polytope</a>
%H A140800 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A140800 For n > 4, a(n) = A086653(n) + 1 = 2^n + 3*n + 1.
%F A140800 G.f.: -(1488*x^7 - 3656*x^6 + 2794*x^5 - 569*x^4 - 58*x^3 + 3*x^2 + 2*x - 1)/((1-x)^2*(1-2*x)). [_Colin Barker_, Sep 05 2012]
%e A140800 a(0) = 1 because the 0-D regular polytope is the point.
%e A140800 a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
%e A140800 a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
%e A140800 a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
%e A140800 a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
%e A140800 For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).
%t A140800 LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* _Georg Fischer_, May 03 2019 *)
%Y A140800 Cf. A000943, A000944, A019503, A053016, A060296, A063924, A063925, A063926, A063927, A065984, A086653, A093478, A093479, A105230, A105231.
%K A140800 easy,nonn
%O A140800 0,2
%A A140800 _Jonathan Vos Post_, Jul 15 2008
%E A140800 a(14)-a(15) corrected by _Georg Fischer_, May 02 2019