

A331451


Triangle read by rows: Take an nsided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of ksided polygons in that figure for k = 3, 4, ..., n.


17



1, 4, 0, 10, 0, 1, 18, 6, 0, 0, 35, 7, 7, 0, 1, 56, 24, 0, 0, 0, 0, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 0, 0, 0, 0, 0, 176, 132, 44, 22, 0, 0, 0, 0, 1, 276, 168, 0, 0, 0, 0, 0, 0, 0, 0, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 0, 0, 0, 0, 0, 0, 0, 0, 0, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET

3,2


COMMENTS

Computed by Scott R. Shannon, Jan 24 2020


LINKS

Table of n, a(n) for n=3..107.
M. Rubinstein, Drawings of A007678 for n=4,5,6,...
Scott R. Shannon, Rows 3 through 45
N. J. A. Sloane, Illustration for row n=9. [9gon with one representative for each type of polygonal cell labeled with its number of sides]


FORMULA

By counting edges in two ways, we have the identity Sum_k k*T(n,k) + n = 2*A135565(n). E.g. for n=7, 3*35+4*7+5*7+6*0+7*1+7 = 182 = 2*A135565(7).


EXAMPLE

A hexagon with all diagonals drawn contains 18 triangles, 6 quadrilaterals, and no pentagons or hexagons, so row 6 is [18, 6, 0, 0].
Triangle begins:
1,
4,0,
10,0,1,
18,6,0,0,
35,7,7,0,1,
56,24,0,0,0,0,
90,36,18,9,0,0,1,
120,...
The row sums are A007678, the first column is A062361.


CROSSREFS

Cf. A007678, A062361.
See A331450 for a version of this triangle in which trailing zeros in the rows have been omitted.
Sequence in context: A098487 A174381 A184363 * A164735 A293933 A345057
Adjacent sequences: A331448 A331449 A331450 * A331452 A331453 A331454


KEYWORD

nonn,tabl


AUTHOR

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020


STATUS

approved



