A236384 Number of non-congruent integer triangles with base length n whose apex lies on or within a space bounded by a semicircle of diameter n.

0, 0, 1, 1, 3, 4, 5, 7, 10, 13, 15, 17, 22, 25, 30, 33, 38, 42, 48, 54, 58, 65, 71, 76, 85, 92, 100, 106, 114, 123, 130, 140, 149, 159, 170, 177, 189, 197, 211, 222, 231, 243, 255, 269, 282, 292, 306, 318, 333, 348, 364, 378, 391, 406, 420, 438, 453, 470, 485

Offset: 1

Author

Frank M Jackson, Jan 24 2014

Keywords

• nonn

Comments

Number of integer-sided obtuse or right (non-acute) triangles with largest side n. - Frank M Jackson, Dec 03 2014

Examples

			a(5)=3 as there are 3 non-congruent integer triangles with base length 5 whose apex lies on or within the space bounded by the semicircle of diameter 5. The integer triples are (2,4,5), (3,3,5), (3,4,5).
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002620, A247588.

Programs

GeoGebra

c = Slider(1, 20, 1);
L = Flatten(Sequence(Sequence(((a^2+c^2-(c-a+k)^2)/(2c),((a+(c-a+k)+c)(a+(c-a+k)-c)(a-(c-a+k)+c)(-a+(c-a+k)+c))^(1/2)/(2c)),a,k,(c+k)/2),k,1,c));
C = {Circle((c/2,0),c/2)};
a_n = CountIf(IsInRegion((x(A),y(A)),Element(C,1)),A,L);
# Frank M Jackson, Jan 01 2024

Mathematica

sumtriangles[c_] := (n=0; Do[If[a^2+b^2<=c^2, n++], {b, 1, c}, {a, c-b+1, b}]; n); Table[sumtriangles[m], {m, 1, 200}]

PARI

a(n)=sum(a=2,n,sum(b=max(a,n+1-a),n,a^2+b^2<=n^2)) \\ Charles R Greathouse IV, Mar 26 2014

PARI

a(n)=sum(a=2,n,max(min(sqrtint(n^2-a^2),n)-max(a,n+1-a)+1,0)) \\ Charles R Greathouse IV, Mar 26 2014

Formula

a(n) = A002620(n+1)-A247588(n). - Frank M Jackson, Dec 03 2014