A096004 - OEIS

%I #32 May 27 2020 21:04:39

%S 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,

%T 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,

%U 4,9,9,11,5,5,8,9,4,9,9,13,5,7,9,8,5,8,9,12

%N Number of convex triangular polyominoes containing n cells.

%C 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.

%C 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

%H Rainer Rosenthal, <a href="/A096004/b096004.txt">Table of n, a(n) for n = 1..5200</a>

%H Peter Kagey, <a href="/A096004/a096004.txt">Examples for a(1)-a(30)</a>.

%F a(n) >= sqrt(n)/3. - _Baohua Tian_, Apr 21 2020

%e a(8)=3 because there are 3 ways to compose a convex polygon of 8 equilateral triangles with side 1:

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%p 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

%Y Cf. A000577, A014529.

%K easy,nonn

%O 1,4

%A _Paul Boddington_, Jul 27 2004

%E a(83) and a(84) corrected by _Rainer Rosenthal_, Sep 20 2017