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 A286430 Least volume of water to surround the largest possible island in a number square. 2
 0, 0, 0, 0, 36, 78, 136, 210, 300, 406, 528, 666, 820, 990, 1176, 1378, 1596, 1830, 2080, 2346, 2628, 2926, 3240, 3570, 3916, 4278, 4656, 5050, 5460, 5886, 6328, 6786, 7260 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The water retention model for mathematical surfaces showed that a random two-level system will contain more water than a random 3-level system when the size of the square is > 52 X 52. It has also been the subject of Zimmermann's programming contest in 2010 and a Wikipedia page as noted below. The number square is a simple environment in which to explore the interaction of volumes, heights, and areas of lakes, ponds, islands, and spillways in the square. A number square contains the numbers for 1 to n^2 without repeats in an n X n square. This sequence is 4*A000217 for a(n) > 8. LINKS Table of n, a(n) for n=0..32. Craig Knecht, 3D graphic. Craig Knecht, Least volume of water to surround the largest island in a number square. Craig Knecht, Number range for each cell type. Wikipedia, Water retention on mathematical surfaces FORMULA Conjectures from Colin Barker, Jan 20 2018: (Start) G.f.: 2*x^4*(18 - 15*x + 5*x^2) / (1 - x)^3. a(n) = 28 - 30*n + 8*n^2 for n>3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End) EXAMPLE For this 5 X 5 square the numbers 1 to 25 are used without repeats. The values 1 through 8 form the moat. The spillway value is 9. The volume of water retained is 36 units. ( 24 23 22 21 20) ( 18 1 2 3 19) ( 17 8 25 4 9) ( 16 7 6 5 15) ( 14 13 12 11 10) CROSSREFS Cf. A054247, A201126, A268311. Sequence in context: A255094 A261284 A111163 * A279901 A335276 A271629 Adjacent sequences: A286427 A286428 A286429 * A286431 A286432 A286433 KEYWORD nonn AUTHOR Craig Knecht, May 09 2017 STATUS approved

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Last modified February 26 09:21 EST 2024. Contains 370338 sequences. (Running on oeis4.)