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 A336660 a(n) is the maximal number of 1 X 1 squares in an arrangement of n squares, from 1 X 1 to n X n. 2
 1, 1, 4, 7, 12, 17, 24, 31, 42, 50, 65 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Terms were calculated by hand by the author. Terms should be verified. On the infinite square grid we can draw every square anywhere, on any of the four quadrants or on several of them. Note that distinct arrangements could give the same result for some values of n. a(7) >= 24. - Hugo Pfoertner, based on data from Hermann Jurksch, Aug 02 2020 From Hugo Pfoertner, Aug 28 2020: (Start) Hermann Jurksch (private communication) has provided a proof that a(n) <= floor((n^2 + n + 1)*(3/5)). I myself conjecture that a(n) <= floor((n^2 - n)*(3/5)) for n >= 4. Lower bounds based on numerical results for the terms a(12)-a(15) are 76, 88, 105, 118. (End) REFERENCES Rodolfo Kurchan, Mesmerizing Math Puzzles, Sterling Publications, (2000), problem 87, Square Stamps, p. 57. LINKS Rodolfo Kurchan, Problem 578, Puzzle Fun, December 1997 Rodolfo Kurchan, Solutions to problem 578, Puzzle Fun, December 1998 EXAMPLE From Omar E. Pol, Jul 30 2020: (Start) For n = 1 we draw a 1 X 1 square, so a(1) = 1. For n = 2 we draw a 2 X 2 square and then a 1 X 1 square, both with a vertex at the point (0,0) as shown below:     _ _    |_  |    |_|_| . We can see only one 1 X 1 square in the arrangement, so a(2) = 1. For n = 3 first we draw a 3 X 3 square with a vertex at the point (0,0) and then we draw a 2 X 2 square centered at the point (3,3) as shown below: .       _ _     _ _|_  |    |   |_|_|    |     |    |_ _ _| . Note that on the cell (3,3) we have formed a 1 X 1 square. Finally we draw a 1 X 1 square on the cell (4,4) as shown below: .       _ _     _ _|_|_| <-- cell (4,4)    |   |_|_|    |     |    |_ _ _|. Note that on the cells (3,4) and (4,3) we have formed two new 1 X 1 squares, hence the total number of 1 X 1 squares in the arrangement is equal to 4, so a(3) = 4. (End) CROSSREFS Cf. A336659 (another version), A337515 (with links to list of solutions and illustrations). Sequence in context: A246399 A276222 A074148 * A310792 A178907 A265431 Adjacent sequences:  A336657 A336658 A336659 * A336661 A336662 A336663 KEYWORD nonn,hard,more AUTHOR Rodolfo Kurchan, Jul 28 2020 EXTENSIONS a(7)-a(11) from Hermann Jurksch and Hugo Pfoertner, Aug 28 2020 STATUS approved

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Last modified October 27 17:15 EDT 2020. Contains 338035 sequences. (Running on oeis4.)