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 A168339 a(n) is the least number of squares needed to form n edge-disjoint 1 X 1 holes inside a rectangle of squares with a solid border. 7
 8, 13, 17, 20, 20, 24, 28, 27, 31, 32, 34, 36, 36, 40, 41, 44, 46, 45, 51, 50, 51, 55, 54, 56, 56, 62, 61, 62, 67, 66, 68, 67, 71, 74, 73, 74, 80, 79, 78, 80, 80, 84, 87, 86, 87, 89, 93, 92, 94, 93, 99, 98, 100, 100, 101, 104, 108, 107, 106, 108, 108, 114, 113, 116, 115, 116 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let the resulting rectangle have size k X l, with a(n) = kl-n. If the border is removed, we have a rectangle of size x X y, where x = k-2, y = l-2. The inner rectangle contains b squares, where b = xy-n = a(n) - 2x - 2y - 4. For k, l, x, y, b see A143082, A145288, A161357, A161358, A161359. The problem therefore reduces to choosing positive numbers x and y such that floor((xy+1)/2) >= n and (x+2)*(y+2) is minimized. The largest number of 1 X 1 holes which can put inside an r X r square with a solid border is (for r>=1) 0,0,1,2,5,8,13,18,25,..., which is essentially A000982. LINKS EXAMPLE a(1)=8 because to create a rectangle with one hole inside, 8 squares are needed, as follows: .HHH .H H .HHH a(2)=13 because to create a rectangle with two holes inside, 13 squares are needed, as follows: .HHHHH .H H H .HHHHH a(3)=17 because to create a rectangle with three holes inside, 17 squares are needed, as follows: .HHHHH .H H H .HH HH .HHHHH a(4)=20 because to create a rectangle with four holes inside, 20 squares are needed, as follows: .HHHHHH .H H HH .HH H H .HHHHHH a(5)=20 because to create a rectangle with 5 holes inside, 20 squares are needed, as follows: .HHHHH .H H H .HH HH .H H H .HHHHH MATHEMATICA A168339 = Reap[For[holes = 1, holes <= 10000, holes++, cost = 2^30; For[n = 1, cost > 2*n + 6, n++, For[m = 1, m <= n, m++, maxholes = n*m - Quotient[n*m, 2]; If[maxholes < holes, Continue[]]; c = 2*(n + m + 2) + n*m - holes; If[c < cost, cost = c; dimn = n; dimm = m]]]; Sow[cost]; If[cost > 120, Break[]]]][[2, 1]] (* Jean-François Alcover, May 15 2017, translated from R. H. Hardin's program *) PROG (C program from R. H. Hardin) #include main() { int holes, cost, c, n, m, maxholes, dimn, dimm; for(holes=1; holes<=10000; holes++) { cost=(1<<30); for(n=1; cost>2*n+6; n++) { for(m=1; m<=n; m++) { maxholes=n*m-((n*m)/2); if(maxholes

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Last modified August 4 15:50 EDT 2020. Contains 336202 sequences. (Running on oeis4.)