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

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

4,6

Comments

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.

Links

Table of n, a(n) for n=4..85.

Vladimir Letsko, Table of rows of a(n)

Example

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;

Maple

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;

Crossrefs

Row lengths are in A023536.

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

Sequence in context: A293497 A278164 A328480 * A053645 A212598 A362190

Adjacent sequences: A279678 A279679 A279680 * A279682 A279683 A279684

Keyword

nonn,tabf

Author

Vladimir Letsko, Dec 16 2016

Status

approved