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A331451 Triangle read by rows: Take an n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided 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 (list; table; graph; refs; listen; history; text; internal format)
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. [9-gon 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

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Last modified September 17 14:07 EDT 2021. Contains 347478 sequences. (Running on oeis4.)