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

1,5

Comments

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

Links

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

Formula

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

Example

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).

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}]

Prog

(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
(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

Crossrefs

Cf. A002620, A247588.

Sequence in context: A030502 A201025 A288451 * A351741 A073957 A309916

Adjacent sequences: A236381 A236382 A236383 * A236385 A236386 A236387

Keyword

nonn

Author

Frank M Jackson, Jan 24 2014

Status

approved