A024156 -id:A024156 - cp's OEIS frontend

A070101 Number of obtuse integer triangles with perimeter n.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 3, 2, 3, 3, 5, 3, 7, 4, 8, 5, 9, 7, 10, 8, 11, 9, 14, 11, 16, 12, 18, 14, 19, 17, 21, 18, 23, 21, 27, 22, 30, 24, 32, 27, 34, 30, 37, 33, 40, 35, 44, 37, 47, 40, 50, 44, 53, 49, 56, 52, 60, 55, 64, 57, 68

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is obtuse iff A070085(k) < 0.

			For n=14 there are A005044(14)=4 integer triangles: [2,6,6], [3,5,6], [4,4,6] and [4,5,5]; two of them are obtuse, as 3^2+5^2<36=6^2 and 4^2+4^2<36=6^2, therefore a(14)=2.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070102, A070103, A070127.

Cf. A005044, A024155, A070093.

Table[Sum[Sum[(1 - Sign[Floor[(i^2 + k^2)/(n - i - k)^2]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

a(n) = A005044(n) - A070093(n) - A024155(n).
a(n) = A024156(n) + A070106(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)}
(1-sign(floor((i^2 + k^2)/(n-i-k)^2))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

A070105 Number of integer triangles with perimeter n and prime side lengths which are obtuse and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 5, 0, 5, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

a(n) = 0 if n is even. - Robert Israel, Jul 26 2024

Robert Israel, Table of n, a(n) for n = 1..10000
R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A005044, A070088, A070090, A070103, A024156, A070132.

f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  if n::even then return 0 fi;
  for a from 1 to n/3 by 2 do
    if not isprime(a) then next fi;
    bmin:= max(a+1,(n+1)/2-a); if bmin::even then bmin:= bmin+1 fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> isprime(b) and isprime(n-a-b), [seq(b,b=bmin .. bmax,2)]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

A070106 Number of integer triangles with perimeter n which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 4, 4, 3, 4, 4

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n)=A070101(n)-A024156(n); a(n)=A059169(n)-A070098(n).

			For n=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) is also isosceles; therefore a(11)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070107, A070108, A070133.

A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 1, 4, 3, 6, 2, 7, 4, 8, 4, 8, 6, 10, 6, 12, 8, 14, 8, 16, 11, 18, 11, 17, 14, 21, 12, 25, 18, 25, 15, 30, 19, 32, 20, 32, 25, 38, 23, 40, 28, 41, 28, 47, 31, 51, 34, 46, 40, 55, 35, 61, 44, 58, 41, 68

Offset: 1

Reinhard Zumkeller, May 05 2002

Robert Israel, Table of n, a(n) for n = 1..10000
R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A051493, A005044, A070102, A024156, A070084, A070131.

f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  for a from 1 to n/3 do
    if n::even then bmin:= max(a+1,n/2-a+1) else bmin:= max(a+1,(n+1)/2-a) fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> igcd(a,b,n-a-b) = 1, [$bmin .. bmax]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

A070130 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle.

Original entry on oeis.org

8, 13, 20, 21, 25, 29, 30, 36, 37, 41, 42, 44, 49, 50, 56, 57, 59, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 87, 89, 96, 97, 99, 100, 101, 102, 105, 110, 111, 113, 115, 122, 123, 125, 126, 127, 128, 130, 131, 138, 139, 141, 143

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(20)=67: [A070080(67), A070081(67), A070082(67)]=[4<7<9], A070085(67)=4^2+7^2-9^2=16+49-81=-16<0.

R. Zumkeller, Integer-sided triangles

Cf. A024156, A070132, A070131, A070112, A070130.

cp's OEIS Frontend

A070101 Number of obtuse integer triangles with perimeter n.

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A070105 Number of integer triangles with perimeter n and prime side lengths which are obtuse and scalene.

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A070106 Number of integer triangles with perimeter n which are obtuse and isosceles.

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A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

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Maple

A070130 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle.

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