A279681 - OEIS

%I #20 Dec 21 2016 13:57:27

%S 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,

%T 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,

%U 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28

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

%C 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.

%H Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:table_for_a279681">Table of rows of a(n)</a>

%e Triangle begins:

%e 4  | 0;

%e 5  | 0, 1;

%e 6  | 0, 1, 2, 3;

%e 7  | 0, 1, 2, 3, 4, 5, 6;

%e 8  | 0, 2, 3, 4, 5, 6, 7,  8,  9, 10;

%e 9  | 0, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15;

%e 10 | 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21;

%p 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);

%p dM:=(n, k)->simplify((n-1)*n/2-2*n-k+3-(n-k)*(n-k-3)/2);

%p Dv:=proc(n) local k, DD; DD:={0}:for k from 2 to n/2-1 do

%p DD:=DD union {seq(i, i=dm(n, k)..dM(n, k))} od:

%p DD:=DD union {seq(i, i=dm(n, k-1)..(n-3)*(n-4)/2)}:

%p DD end;

%Y Row lengths are in A023536.

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

%K nonn,tabf

%O 4,6

%A _Vladimir Letsko_, Dec 16 2016