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A186053
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Smallest perimeter among all sets of nonnegative integers whose volume (sum) is n.
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8
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0, 1, 2, 2, 4, 5, 3, 6, 7, 6, 4, 8, 8, 9, 7, 5, 10, 11, 9, 10, 8, 6, 11, 12, 13, 10, 11, 9, 7, 14, 12, 13, 14, 11, 12, 10, 8, 16, 16, 13, 14, 15, 12, 13, 11, 9, 16, 17, 17, 14, 15, 16, 13, 14, 12, 10, 16, 17, 18, 18, 15, 16, 17, 14, 15, 13, 11, 22, 17, 18, 19, 19, 16, 17
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OFFSET
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0,3
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COMMENTS
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The volume and perimeter of a set S of nonnegative integers are introduced in the reference. The volume is defined simply as the sum of the elements of S, and the perimeter is defined as the sum of the elements of S whose predecessor and successor are not both in S.
For all partitions into distinct parts (with first part 0 implied), the perimeter of a set is the sum of parts p such that not both of p-1 and p+1 are in the partition. The partition with smallest perimeter is sometimes, but not often unique. For example, there are three partitions of volume 37 achieving the minimal perimeter 16, namely [ 0 1 2 3 4 5 6 7 x 9 ], [ x 2 x 5 6 7 8 9 ], and [0 x 2 x 5 6 7 8 9 ] (where x is for one or more skipped parts), but there is only one partition of 36 with minimal perimeter 8, namely [ 0 1 2 3 4 5 6 7 8 ]. [Joerg Arndt, Jun 03 2013]
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LINKS
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FORMULA
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If t(n) is a triangular number t(n)=n*(n+1)/2, then a(t(n))=n.
a(n) = A002024(n) + A182298(A025581(n)) unless n is one of the 114 known counterexamples listed in A182246. This allows the n-th term to be calculated in order log(log(n)) time using constant memory. The first n terms can be calculated in order n time (using order n memory). [Result and details in above listed paper by Patrick Devlin (2012).]
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EXAMPLE
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For n=8, the set S={0,1,2,5} has sum 8 and the perimeter 7 (the sum of 2 and 5). No other set of volume 8 has a smaller perimeter, so a(8)=7.
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0 and i<>0, `if`(t>1, i+1, 0),
`if`(i<0, 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):
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MATHEMATICA
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notBoth[lst_List] := Select[lst, !MemberQ[lst, #-1] || !MemberQ[lst, #+1]&]; Table[s=Select[IntegerPartitions[n], Length[#]==Length[Union[#]]&]; s=Append[#, 0]&/@s; Min[Table[Plus@@notBoth[i], {i, s}]], {n, 40}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 && i != 0, If[t > 1, i+1, 0], If[ i<0, 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, Aug 29 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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