A247588
Number of integer-sided acute triangles with largest side n.
Original entry on oeis.org
1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52, 56, 63, 67, 73, 80, 84, 90, 96, 104, 111, 117, 126, 132, 140, 147, 154, 165, 172, 183, 189, 198, 210, 219, 229, 237, 247, 260, 270, 282, 292, 302
Offset: 1
a(3) = 3 because there are 3 integer-sided acute triangles with largest side 3: (1,3,3); (2,3,3); (3,3,3).
-
tr_a:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d>0 then t:=t+1 fi
od od;
t; end;
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a[ n_] := Length @ FindInstance[ n >= b >= a >= 1 && n < b + a && n^2 < b^2 + a^2, {a, b}, Integers, 10^9]; (* Michael Somos, May 24 2015 *)
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a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014
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{a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */
A247586
Number of acute triangles with integer sides less than or equal to n.
Original entry on oeis.org
1, 3, 6, 11, 17, 25, 36, 49, 64, 81, 102, 127, 154, 185, 219, 258, 301, 349, 401, 457, 520, 587, 660, 740, 824, 914, 1010, 1114, 1225, 1342, 1468, 1600, 1740, 1887, 2041, 2206, 2378, 2561, 2750, 2948
Offset: 1
a(2) = 3 because there are 3 acute triangles with integer sides less than or equal to 2: (1,1,1); (1,2,2); (2,2,2).
-
tr_a:=proc(n) local a,b,c,t,d;t:=0:
for a to n do
for b from a to n do
for c from b to min(a+b-1,n) do
d:=a^2+b^2-c^2:
if d>0 then t:=t+1 fi
od od od;
[n,t]; end;
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a[n_] := Module[{a, b, c, d, t = 0}, Do[d = a^2 + b^2 - c^2; If[d>0, t++], {a, n}, {b, a, n}, {c, b, Min[a+b-1, n]}]; t]; Array[a, 40] (* Jean-François Alcover, Jun 19 2019, from Maple *)
-
import itertools
def A247586(n):
I = itertools.combinations_with_replacement(range(1,n+1),3)
F = filter(lambda c: c[0]**2 + c[1]**2 > c[2]**2, I)
return len(list(F))
print([A247586(n) for n in range(41)]) # Peter Luschny, Sep 22 2014
A306674
Number of distinct non-similar obtuse triangles with integer sides and length of largest side <= n.
Original entry on oeis.org
0, 0, 1, 2, 5, 9, 14, 21, 31, 44, 59, 76, 98, 123, 153, 186, 224, 266, 314, 368, 426, 491, 562, 638, 723, 815, 915, 1021, 1135, 1258, 1388, 1528, 1677, 1836, 2006, 2183, 2372, 2569, 2780, 3002, 3233, 3476, 3731, 4000, 4282, 4574, 4880, 5198, 5531, 5879
Offset: 1
For n=6, there are 9 integer-sided obtuse triangles with largest side <= n. These have sides {a, b, c} = {2, 2, 3}, {2, 3, 4}, {2, 4, 5}, {2, 5, 6}, {3, 3, 5}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 4, 6}. But {4, 4, 6} is similar to {2, 2, 3} and is excluded from the list, so a(6) = 8.
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#nType=1 for acute triangles, nType=2 for obtuse triangles, nType=0 for both triangles
CountTriangles := proc (n, nType := 1)
local aa, oo, a, b, c, tt, lAcute;
aa := {}; oo := {};
for a from n by -1 to 1 do for b from a by -1 to 1 do for c from b by -1 to 1 do
if a < b+c and abs(b-c) < a and b < c+a and abs(c-a) < b and c < a+b and abs(a-b) < c and gcd(a, gcd(b, c)) = 1 then
lAcute := evalb(0 < b^2+c^2-a^2);
tt := sort([a, b, c]);
if lAcute then aa := {op(aa), tt} else oo := {op(oo), tt} end if
end if
end do end do end do;
return sort(`if`(nType = 1, aa, `if`(nType=2,oo,`union`(aa,oo))))
end proc
A247589
Number of integer-sided obtuse triangles with largest side n.
Original entry on oeis.org
0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 17, 21, 25, 29, 33, 37, 42, 48, 53, 58, 65, 71, 76, 83, 91, 100, 106, 113, 122, 130, 140, 149, 158, 169, 177, 188, 197, 210, 221, 230, 243, 255, 269, 281, 292, 306, 318, 333, 346
Offset: 1
a(5) = 2 because there are 2 integer-sided acute triangles with largest side 5: (2,4,5); (3,3,5).
-
tr_o:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d<0 then t:=t+1 fi
od od;
t; end;
Showing 1-4 of 4 results.