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 A331757 Number of edges in a figure made up of a row of n adjacent congruent rectangles upon drawing diagonals of all possible rectangles. 15
 8, 28, 80, 178, 372, 654, 1124, 1782, 2724, 3914, 5580, 7626, 10352, 13590, 17540, 22210, 28040, 34670, 42760, 51962, 62612, 74494, 88508, 104042, 121912, 141534, 163664, 187942, 215636, 245490, 279260, 316022, 356456, 399898, 447612, 498698, 555352 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918. FORMULA a(n) = (2*n + 2 + 3*A324042(n) + 4*A324043(n))/2 [Corrected by Chai Wah Wu, Aug 16 2021] For n > 1, a(n) = 2*(n*(n+3) + Sum_{i=2..floor(n/2)} (n+1-i)*(n+1+i)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i)). - Chai Wah Wu, Aug 16 2021 MATHEMATICA Table[n^2 + 4n + 1 + Sum[Sum[(2 * Boole[GCD[i, j] == 1] - Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}], {n, 1, 37}] (* Joshua Oliver, Feb 05 2020 *) PROG (Python) from sympy import totient def A331757(n): return 8 if n == 1 else 2*(n*(n+3) + sum(totient(i)*(n+1-i)*(n+1+i) for i in range(2, n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1, n+1))) # Chai Wah Wu, Aug 16 2021 CROSSREFS A306302 gives number of regions in the figure. This is column 1 of A331454. Cf. A324042, A324043. Sequence in context: A134638 A293289 A305638 * A130129 A317032 A229713 Adjacent sequences: A331754 A331755 A331756 * A331758 A331759 A331760 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 04 2020 STATUS approved

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Last modified December 6 18:41 EST 2023. Contains 367614 sequences. (Running on oeis4.)