A227538 Smallest k such that a partition of n into distinct parts with perimeter k exists.

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

Offset: 0

Author

Alois P. Heinz, Jul 16 2013

Keywords

• nonn
• look
• hear

Comments

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A227344, A186053 (smallest perimeter among all sets of nonnegative integers).

Programs

Maple

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);

Mathematica

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 *)

Formula

a(n) = min { k : A227344(n,k) > 0 }.

a(A000217(n)) = n+1 for n>1.