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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 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.)