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A324042 Number of triangle regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles. 2
4, 14, 32, 70, 124, 226, 360, 566, 820, 1218, 1696, 2310, 3020, 4018, 5160, 6590, 8196, 10218, 12464, 15110, 18012, 21650, 25624, 30142, 35028, 40954, 47344, 54558, 62284, 71034, 80360, 90806, 101892, 114770, 128416, 143286, 158972, 176914, 195816, 216350, 237908, 261546, 286304, 313102, 341100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A row of n adjacent congruent rectangles can only be divided into triangles or quadrilaterals when drawing diagonals. Proof is given in Alekseyev et al. (2015) under the transformation described in A306302.

LINKS

Table of n, a(n) for n=1..45.

Jinyuan Wang, Illustration for n = 1, 2, 3, 4, 5

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165.

Robert Israel, Maple program

FORMULA

a(n) = A177719(n+1) + 2*(n+1) = 2 * ( (n+1)*n + Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) ). - Max Alekseyev, Jul 08 2019

a(n) = A306302(n) - A324043(n).

EXAMPLE

There are k adjacent congruent rectangles, number of triangle regions in the j-th rectangle:

k\j|  1   2   3   4   5   6   7  ...

---+--------------------------------

1  |  4,  0,  0,  0,  0,  0,  0, ...

2  |  7,  7,  0,  0,  0,  0,  0, ...

3  |  9, 15,  9,  0,  0,  0,  0, ...

4  | 11, 24, 24, 11,  0,  0,  0, ...

5  | 13, 30, 38, 30, 13,  0,  0, ...

6  | 15, 38, 60, 60, 38, 15,  0, ...

7  | 17, 44, 76, 86, 76, 44, 17, ...

...

a(4) = 11 + 24 + 24 + 11 = 70.

MAPLE

See link.

PROG

(PARI) { A324042(n) = 2*((n+1)*n + sum(i=1, n, sum(j=1, n, (gcd(i, j)==2)*(n+1-i)*(n+1-j))) ); } \\ Max Alekseyev, Jul 08 2019

CROSSREFS

Cf. A177719, A306302, A324043.

Sequence in context: A023627 A023649 A323723 * A099590 A078432 A124786

Adjacent sequences:  A324039 A324040 A324041 * A324043 A324044 A324045

KEYWORD

nonn

AUTHOR

Jinyuan Wang, May 01 2019

EXTENSIONS

a(8)..a(23) from Robert Israel, Jul 07 2019

Terms a(24) onward from Max Alekseyev, Jul 08 2019

STATUS

approved

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Last modified January 28 03:28 EST 2020. Contains 331314 sequences. (Running on oeis4.)