A348149 Variation of the Barnyard sequence A347581: a(n) is the minimum number of unit-length line segments required to enclose areas of 1 through n on a square grid when the number of segments is minimized as each area of incrementing size, starting at 1, is added.

4, 9, 14, 20, 26, 33, 40, 48, 55, 64

Offset

1,1

Comments

In this variation of A347581 the areas must be added in the order of their sizes, from 1 through n, and as each area is added the minimum possible number of line segments must be used. This forces, for example, the first three areas of size 1, 2 and 3 to form a 2 X 3 block and thus they can never appear in any other arrangement in the final area. This is also true for n up to at least 9 due to the restriction of maximizing the usable edges for the next area. This leads to a(8) and a(10) containing one more line segment than the optimal solutions of A347581.

Links

Table of n, a(n) for n=1..10.

Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2015.

Example

Examples of n = 1 to n = 10 are given below. Note that for a(3) the configuration could also consist of the area of size 1 sitting above the area of size 2 with the area of size 3 forming an L-shaped block creating the minimal 2 X 3 block.
.
   __
  |__|  a(1) = 4
   __ __ __
  |__|__ __|  a(2) = 9
   __ __ __
  |__|__ __|  a(3) = 14
  |__ __ __|
   __ __ __
  |__|__ __|
  |__ __ __|  a(4) = 20
  |     |
  |__ __|
   __ __ __
  |__|__ __|__
  |__ __ __|  |  a(5) = 26
  |     |     |
  |__ __|__ __|
   __ __ __
  |__|__ __|__ __ __
  |__ __ __|  |     |  a(6) = 33
  |     |     |     |
  |__ __|__ __|__ __|
         __ __ __ __
   __ __|__         |
  |__|__ __|__ __ __|
  |__ __ __|  |     |  a(7) = 40
  |     |     |     |
  |__ __|__ __|__ __|
         __ __ __ __
        |           |
        |__ __ __ __|
   __ __|__         |
  |__|__ __|__ __ __|  a(8) = 48
  |__ __ __|  |     |
  |     |     |     |
  |__ __|__ __|__ __|
   __ __ __ __ __ __ __
  |        |           |
  |        |__ __ __ __|
  |__ __ __|__         |
     |__|__ __|__ __ __|  a(9) = 55
     |__ __ __|  |     |
     |     |     |     |
     |__ __|__ __|__ __|
      __ __ __ __ __ __ __
     |        |           |
     |        |__ __ __ __|
   __|__ __ __|__         |
  |     |__|__ __|__ __ __|  a(10) = 64
  |     |__ __ __|  |     |
  |     |     |     |     |
  |     |__ __|__ __|__ __|
  |__ __|
.

Crossrefs

Cf. A347581, A001168, A291808, A291809, A328020, A291806, A006983.

Sequence in context: A313154 A313155 A347581 * A023488 A115585 A099980

Adjacent sequences: A348146 A348147 A348148 * A348150 A348151 A348152

Keyword

nonn,more

Author

Scott R. Shannon, Oct 03 2021

Status

approved