A279620 -id:A279620 - cp's OEIS frontend

A318930 RUNS transform of A279620.

1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

N. J. A. Sloane, Sep 09 2018

nonn

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

Cf. A279620.

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.

cp's OEIS Frontend

A318930 RUNS transform of A279620.

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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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