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%I #32 Jan 21 2024 13:16:02
%S 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,
%T 92,100,106,114,123,130,140,149,159,170,177,189,197,211,222,231,243,
%U 255,269,282,292,306,318,333,348,364,378,391,406,420,438,453,470,485
%N 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.
%C Number of integer-sided obtuse or right (non-acute) triangles with largest side n. - _Frank M Jackson_, Dec 03 2014
%H Giovanni Resta, <a href="/A236384/b236384.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A002620(n+1)-A247588(n). - _Frank M Jackson_, Dec 03 2014
%e 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).
%t 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}]
%o (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
%o (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
%o (GeoGebra)
%o c = Slider(1, 20, 1);
%o 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));
%o C = {Circle((c/2,0),c/2)};
%o a_n = CountIf(IsInRegion((x(A),y(A)),Element(C,1)),A,L);
%o # _Frank M Jackson_, Jan 01 2024
%Y Cf. A002620, A247588.
%K nonn
%O 1,5
%A _Frank M Jackson_, Jan 24 2014
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