OFFSET
1,4
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
LINKS
Rainer Rosenthal, Table of n, a(n) for n = 1..5200
Peter Kagey, Examples for a(1)-a(30).
FORMULA
a(n) >= sqrt(n)/3. - Baohua Tian, Apr 21 2020
EXAMPLE
a(8)=3 because there are 3 ways to compose a convex polygon of 8 equilateral triangles with side 1:
.
*---*---*---*---*
/ \ / \ / \ / \ /
*---*---*---*---*
*---*---*
/ \ / \ /
*---*---*
/ \ / \ /
*---*---*
*---*
/ \ / \
*---*---*
/ \ / \ / \
*---*---*---*
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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Boddington, Jul 27 2004
EXTENSIONS
a(83) and a(84) corrected by Rainer Rosenthal, Sep 20 2017
STATUS
approved