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 A279015 Greatest possible number of diagonals of a polyhedron having n faces. 6
 0, 0, 4, 10, 20, 34, 52, 73, 100, 128, 162, 199, 240, 285, 334, 387, 444, 505, 570, 639, 712, 789, 870, 955, 1044, 1137, 1234, 1335, 1440, 1549, 1662, 1779, 1900, 2025, 2154, 2287, 2424, 2565, 2710, 2859, 3012, 3169, 3330, 3495, 3664, 3837, 4014, 4195, 4380, 4569 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,3 COMMENTS Also the greatest possible number of diagonals of a simple polyhedron with n faces. In other words a polyhedron with n faces having the greatest possible number of diagonals must be a simple one. LINKS Colin Barker, Table of n, a(n) for n = 4..1000 D. Bremner, V. Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197. Vladimir Letsko, Mathematical Marathon, Problem 219 (in Russian) Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 2*n^2 - 21*n + 64 for n=12 or n>=14. From Colin Barker, Dec 05 2016: (Start) a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>8. G.f.: x^6*(4 - 2*x + 2*x^2 - x^5 + 3*x^6 - 5*x^7 + 5*x^8 - 3*x^9 + x^10) / (1 - x)^3. (End) EXAMPLE a(6)=4 because 6 is the greatest possible number of diagonals of a hexahedron. MAPLE F:=n->piecewise(4<=n and n<=5, 0, 6<=n and n<=10, 2*n^2-20*n+52, n=11, 73, n=13, 128, n=12 or n>=14, 2*n^2-21*n+64); MATHEMATICA Drop[#, 4] &@ CoefficientList[Series[x^6*(4 - 2 x + 2 x^2 - x^5 + 3 x^6 - 5 x^7 + 5 x^8 - 3 x^9 + x^10)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Dec 05 2016 *) PROG (PARI) concat(vector(2), Vec(x^6*(4 - 2*x + 2*x^2 - x^5 + 3*x^6 - 5*x^7 + 5*x^8 - 3*x^9 + x^10) / (1 - x)^3 + O(x^30))) \\ Colin Barker, Dec 05 2016 CROSSREFS Cf. A000944, A279019, A279022. Sequence in context: A099589 A008141 A119651 * A005893 A301034 A301030 Adjacent sequences:  A279012 A279013 A279014 * A279016 A279017 A279018 KEYWORD nonn,easy AUTHOR Vladimir Letsko, Dec 03 2016 STATUS approved

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Last modified July 4 19:28 EDT 2022. Contains 355084 sequences. (Running on oeis4.)