A096004 Number of convex triangular polyominoes containing n cells.

1, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 5, 2, 3, 3, 4, 2, 4, 4, 6, 3, 3, 4, 5, 2, 5, 5, 7, 3, 4, 5, 6, 3, 5, 5, 8, 3, 4, 5, 6, 4, 7, 7, 9, 4, 5, 5, 7, 3, 7, 8, 9, 3, 5, 7, 8, 4, 8, 8, 11, 4, 5, 7, 8, 4, 9, 9, 11, 5, 5, 8, 9, 4, 9, 9, 13, 5, 7, 9, 8, 5, 8, 9, 12

Offset: 1

Author

Paul Boddington, Jul 27 2004

Keywords

• easy
• nonn
• changed

Comments

The main sequence on triangular polyominoes is A000577. The convexity condition makes enumeration easy as a convex triangular polyomino has at most 6 sides. It is simple to prove that a(n) is also the number of 4-tuples (p,b,c,d) of nonnegative integers satisfying b<=c<=d, b+c+d<=p, n=p^2-b^2-c^2-d^2.

For n = A014529(k) there are a(n) many polygons. At least one of them can be tiled with k equilateral triangles. - Rainer Rosenthal, Sep 20 2017

Examples

			a(8)=3 because there are 3 ways to compose a convex polygon of 8 equilateral triangles with side 1:
.
    *---*---*---*---*
   / \ / \ / \ / \ /
  *---*---*---*---*
      *---*---*
     / \ / \ /
    *---*---*
   / \ / \ /
  *---*---*
      *---*
     / \ / \
    *---*---*
   / \ / \ / \
  *---*---*---*
		

Links

• Rainer Rosenthal, Table of n, a(n) for n = 1..5200
• Peter Kagey, Examples for a(1)-a(30).
• Walter Trump, Number of Convex Polygons of a Given Perimeter or Area on the Triangular Lattice, 2025.

Crossrefs

Cf. A000577, A014529.

Programs

Maple

a:=proc(n) local x,p,d,c,b; x:=0; for p from 0 to ceil((n+1)/2) do; for d from 0 to p do; for c from 0 to min(d,p-d) do; for b from 0 to min(c,p-c-d) do; if p^2-b^2-c^2-d^2=n then x:=x+1 fi; od; od; od; od; x; end; # corrected by Rainer Rosenthal, Sep 20 2017

Formula

a(n) >= sqrt(n)/3. - Baohua Tian, Apr 21 2020

Extensions

a(83) and a(84) corrected by Rainer Rosenthal, Sep 20 2017