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A227538
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Smallest k such that a partition of n into distinct parts with perimeter k exists.
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3
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0, 1, 2, 3, 4, 5, 4, 7, 8, 6, 5, 9, 8, 9, 7, 6, 11, 12, 9, 10, 8, 7, 11, 12, 13, 10, 11, 9, 8, 15, 12, 13, 14, 11, 12, 10, 9, 16, 17, 13, 14, 15, 12, 13, 11, 10, 16, 17, 18, 14, 15, 16, 13, 14, 12, 11, 16, 17, 18, 19, 15, 16, 17, 14, 15, 13, 12, 22, 17, 18, 19
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OFFSET
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0,3
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COMMENTS
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The perimeter is the sum of all parts having less than two neighbors.
a(n) is also the smallest perimeter among all sets of positive integers whose volume (sum) is n. - Patrick Devlin, Jul 23 2013
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LINKS
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FORMULA
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a(n) = min { k : A227344(n,k) > 0 }.
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EXAMPLE
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a(0) = 0: the empty partition [] has perimeter 0.
a(1) = 1: [1] has perimeter 1.
a(3) = 3: [1,2], [3] have perimeter 3.
a(6) = 4: [1,2,3] has perimeter 4.
a(7) = 7: [1,2,4], [3,4], [2,5], [1,6], [7] have perimeter 7; no partition of 7 into distinct parts has a smaller perimeter.
a(10) = 5: [1,2,3,4] has perimeter 5.
a(15) = 6: [1,2,3,4,5] has perimeter 6.
a(29) = 15: [1,2,3,4,5,6,8] has perimeter 1+6+8 = 15.
a(30) = 12: [4,5,6,7,8] has perimeter 12.
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, i+1, 0),
`if`(i<1, infinity, min(`if`(t>1, i+1, 0)+b(n, i-1, iquo(t, 2)),
`if`(i>n, NULL, `if`(t=2, i+1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>1, i+1, 0], If[i<1, Infinity, Min[If[t>1, i+1, 0] + b[n, i-1, Quotient[t, 2]], If[i>n, Infinity, If[t == 2, i+1, 0] + b[n-i, i-1, Quotient[t, 2]+2]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)
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CROSSREFS
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Cf. A227344, A186053 (smallest perimeter among all sets of nonnegative integers).
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KEYWORD
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AUTHOR
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STATUS
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approved
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