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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.
4
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
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
Sequence in context: A030502 A201025 A288451 * A351741 A073957 A309916
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jan 24 2014
STATUS
approved