A279015 -id:A279015 - cp's OEIS frontend

A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset: 4

Vladimir Letsko, Dec 03 2016

Obviously, a pyramid has no diagonals. Hence minimum of diagonals of an arbitrary convex polyhedron having n faces is equal to 0.
Minimum number of diagonals among simple convex polyhedra having n faces is obtained from a polyhedron with two triangular faces, n-4 quadrangular faces and two (n-1)-sided faces. A polyhedron having 3 triangular faces, 3 pentagonal faces and 1 hexagonal face gives another example of a simple convex polyhedron with the least possible number of diagonals for n = 7. A polyhedron having 4 triangular faces and 4 hexagonal faces gives a similar example for n = 8.
Essentially the same as A103505 and A002378. - R. J. Mathar, Dec 05 2016

Colin Barker, Table of n, a(n) for n = 4..1000
David Bremner and Victor Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000944, A002378, A279015, A279022, A007531.

Table[(n-4)(n-5),{n,4,60}] (* or *) LinearRecurrence[{3,-3,1},{0,0,2},60] (* Harvey P. Dale, Sep 23 2019 *)

concat(vector(2), Vec(2*x^6 / (1 - x)^3 + O(x^60))) \\ Colin Barker, Dec 05 2016

a(n) = n^2 - 9*n + 20 = (n-4)*(n-5).
G.f.: -2*x^6/(x-1)^3. - R. J. Mathar, Dec 05 2016
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Colin Barker, Dec 05 2016
E.g.f.: exp(x)*(20 - 8*x + x^2) - x^3/3 - 3*x^2 - 12*x - 20. - Stefano Spezia, Nov 24 2019
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=6} 1/a(n) = 1.
Sum_{n>=6} (-1)^n/a(n) = 2*log(2) - 1. (End)

A279022 Greatest possible number of diagonals of a polyhedron having n edges.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 7, 10, 13, 16, 20, 23, 28, 34, 37, 44, 52, 55, 64, 73, 77, 88, 100, 103, 115, 128, 133

Offset: 8

Vladimir Letsko, Dec 03 2016

Note that a polyhedron with 6 edges (a tetrahedron) has no diagonals and a polyhedron having exactly 7 edges does not exist.
If n = 3k where k > 3 than the maximum number of diagonals is achieved by a simple polyhedron with k+2 faces.
According to the Grünbaum-Motzkin Theorem a(3k) = 2*k^2-13*k+30, for all k>11.
Additionally for all k>11 a(3k+1) <= 2*k^2-13*k+36 and a(3k+2) <= 2*k^2-11*k+27.

1. B. Grünbaum, Convex Polytopes, 2nd edition, Springer, 2003.

B. Grünbaum, T. S. Motzkin, The number of hexagons and the simplicity of geodesics of certain polyhedra , Canadian journal of Mathematics, 15 (1963), pp. 744-751.

Cf. A002840, A279015, A279019.

A279681 Irregular triangle read by rows: possible numbers of diagonals of convex polyhedra having n vertices.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Offset: 4

Vladimir Letsko, Dec 16 2016

Let n>4 denote the number of vertices. The set of possible numbers of diagonals is the union of sets {(k-1)(n-k-4), ..., (k-1)(n-(k+6)/2)}, where 1 <= k <= floor((sqrt(8n-15)-5)/2), and the set {(k-1)(n-k-4), ..., (n-3)(n-4)/2}, where k = floor((sqrt(8n-15)-3)/2). Note that cardinalities of all sets of this union excluding the last one are consecutive triangular numbers.

			Triangle begins:
4  | 0;
5  | 0, 1;
6  | 0, 1, 2, 3;
7  | 0, 1, 2, 3, 4, 5, 6;
8  | 0, 2, 3, 4, 5, 6, 7,  8,  9, 10;
9  | 0, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15;
10 | 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21;

Vladimir Letsko, Table of rows of a(n)

Row lengths are in A023536.

Cf. A000217, A279015, A279019, A279022, A279620, A279647, A279679.

dm:=(n, k)->simplify((n-1)*n/2-(2*n-k-1)-(n-k)*(n-k-3)/2-2*(k-1)-(k+2)*(k-1)/2);
dM:=(n, k)->simplify((n-1)*n/2-2*n-k+3-(n-k)*(n-k-3)/2);
Dv:=proc(n) local k, DD; DD:={0}:for k from 2 to n/2-1 do
DD:=DD union {seq(i, i=dm(n, k)..dM(n, k))} od:
DD:=DD union {seq(i, i=dm(n, k-1)..(n-3)*(n-4)/2)}:
DD end;

A279647 Irregular triangle read by rows: possible numbers of the diagonals of an n-faced convex polyhedron.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 0, 2, 3, 4, 5, 6, 7, 9, 10, 0, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34

Offset: 4

Vladimir Letsko, Dec 16 2016

			Triangle begins:
4 | 0;
5 | 0;
6 | 0, 1, 2, 4;
7 | 0, 2, 3, 4, 5, 6, 7,  9, 10;
8 | 0, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20;

Cf. A279015, A279019, A279022, A279620, A279679, A279681.

A279679 Irregular triangle read by rows: possible numbers of diagonals of an n-edged convex polyhedron.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 3, 4, 2, 3, 4, 5, 0, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 9, 10, 0, 4, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 6

Vladimir Letsko, Dec 16 2016

The beginning of the irregular triangle is:
| 0;
| -
| 0;
| 0, 1;
| 0, 1;
| 1, 2;
| 0, 2, 3, 4;
| 2, 3, 4, 5;
| 0, 3, 4, 5,  6,  7;
| 3, 4, 5, 6,  7,  8,  9, 10;
| 0, 4, 6, 7,  8,  9, 10, 11, 12, 13;
| 4, 5, 7, 8,  9, 10, 11, 12, 13, 14, 15, 16;
| 0, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20;

Cf. A279015, A279019, A279022, A279620, A279647, A279681.

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A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

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A279022 Greatest possible number of diagonals of a polyhedron having n edges.

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A279681 Irregular triangle read by rows: possible numbers of diagonals of convex polyhedra having n vertices.

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A279647 Irregular triangle read by rows: possible numbers of the diagonals of an n-faced convex polyhedron.

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A279679 Irregular triangle read by rows: possible numbers of diagonals of an n-edged convex polyhedron.

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