A336753 -id:A336753 - cp's OEIS frontend

A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

2, 3, 4, 3, 4, 5, 3, 5, 7, 4, 5, 6, 4, 6, 8, 5, 6, 7, 4, 7, 10, 5, 7, 9, 6, 7, 8, 5, 8, 11, 6, 8, 10, 7, 8, 9, 5, 9, 13, 6, 9, 12, 7, 9, 11, 8, 9, 10, 6, 10, 14, 7, 10, 13, 8, 10, 12, 9, 10, 11, 6, 11, 16, 7, 11, 15, 8, 11, 14, 9, 11, 13, 10, 11, 12, 7, 12, 17, 8, 12, 16

Offset: 1

Bernard Schott, Aug 03 2020

The triples are displayed in increasing order of perimeter, and if perimeters coincide then by increasing order of the smallest side, hence, each triple (a, b, c) is in increasing order.
Equivalently: triples of integer-sided triangles such that b = (a+c)/2 with a < c.
As the perimeter of these triangles = 3*b, the triples are also displayed in increasing order of middle side.
When a < b < c are in arithmetic progression with b - a = c - b = x, then 1 <= x <= floor((b-1)/2), hence, there exist for each side b >= 3, floor((b-1)/2) = A004526(b) triangles whose sides a < b < c are in arithmetic progression.
The only right integer-sided triangles such that a < b < c are in arithmetic progression correspond to the Pythagorean triples (3k, 4k, 5k) with k > 0.
There do not exist triangles whose sides a < b < c and angles A < B < C are both in arithmetic progression.
Three geometrical properties about these triangles, even if they are not integer-sided:
   1) tan(A/2) * tan(C/2) = 1/3,
   2) r = h_b/3, where r is the inradius and h_b the length of the altitude through B,
   3) The line (IG) is parallel to side (AC), where I is the incenter and G is the centroid of the triangle.

			The smallest such triangle is (2, 3, 4).
The only triangle with perimeter = 12 corresponds to the Pythagorean triple: (3, 4, 5).
There exist two triangles with perimeter = 15 corresponding to triples (3, 5, 7) and (4, 5, 6).
There exist also two triangles with perimeter = 18 corresponding to triples (4, 6, 8) and (5, 6, 7).
The table begins:
  2, 3, 4;
  3, 4, 5;
  3, 5, 7;
  4, 5, 6;
  4, 6, 8;
  5, 6, 7;
  4, 7, 10;
  5, 7, 9;
  6, 7, 8;

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-288 p. 120, André Desvigne.

Michel Marcus, Table of n, a(n) for n = 1..16650 (Rows 1 to 5550)
Wikipedia, Integer triangle.

Cf. A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), A024164 (number of triangles with perimeter = n), A336755 (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of primitive triangles with perimeter = n).
Cf. A004526 (number of triangles with middle side = b).
Cf. A103605 (similar, with Pythagorean triples).
Cf. A335893 (similar, with A, B, C in arithmetic progression).

for b from 3 to 20 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a,b,c);
end do;
end do;

Block[{nn = 12, a, b, c}, Reap[Do[Do[Sow@ {a, b, 2 b - a}, {a, b - Floor[(b - 1)/2], b - 1}], {b, 3, nn}]][[-1, 1]] ] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

tabf(nn) = {for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); print(a, " ", b, " ", c);););} \\ Michel Marcus, Sep 08 2020

T(n,1) = A336751(n); T(n,2) = A307136(n); T(n,3) = A336753(n).

A336754(n) = T(n,1) + T(n,2) + T(n,3).

A336751 Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 4, 5, 6, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14

Offset: 1

Bernard Schott, Aug 15 2020

nonn

The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeter coincide, then by increasing order of the smallest side. This sequence lists the a's.
Equivalently: smallest side of integer-sided triangles such that b = (a+c)/2 with a < c.
a >= 2 and each side a appears a-1 times but not consecutively.
For each a = 3*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.
This sequence is not increasing a(6) = 5 for triangle with perimeter = 18 and a(7) = 4 for triangle with perimeter = 21. The smallest side is not an increasing function of the perimeter of these triangles.
For the corresponding triples and miscellaneous properties and references, see A336750.

			a = 2 for only the smallest triangle (2, 3, 4).
a = 3 for Pythagorean triple (3, 4, 5) and also for the second triangle (3, 5, 7).

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A336750 (triples), this sequence (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter).

Cf. A335894 (smallest side when triangles angles are in arithmetic progression).

for b from 3 to 30 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a);
end do;
end do;

Flatten[Array[Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* Paolo Xausa, Feb 28 2024 *)

a(n) = A336750(n, 1).

A336754 Perimeters in increasing order of integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

9, 12, 15, 15, 18, 18, 21, 21, 21, 24, 24, 24, 27, 27, 27, 27, 30, 30, 30, 30, 33, 33, 33, 33, 33, 36, 36, 36, 36, 36, 39, 39, 39, 39, 39, 39, 42, 42, 42, 42, 42, 42, 45, 45, 45, 45, 45, 45, 45, 48, 48, 48, 48, 48, 48, 48, 51, 51, 51, 51, 51, 51, 51, 51

Offset: 1

Bernard Schott, Aug 31 2020

nonn

Equivalently: perimeters of integer-sided triangles such that b = (a+c)/2 with a < c.
As perimeter = 3 * middle side, these perimeters p are all multiple of 3, and each term p appears floor((p-3)/6) = A004526((p-3)/3) consecutively.
For each perimeter = 12*k with k>0, there exists one right integer triangle whose triple is (3k, 4k, 5k).
For the corresponding primitive triples, miscellaneous properties and references, see A336750.

			Perimeter = 9 only for the smallest triangle (2, 3, 4).
Perimeter = 12 only for Pythagorean triple (3, 4, 5).
Perimeter = 15 for the two triples (3, 5, 7) and (4, 5, 6).

V. Lespinard and R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A004526.
Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), this sequence (perimeter), A024164 (number of such triangles whose perimeter = n), A336755 (primitive triples).
Cf. A335897 (perimeters when angles A, B and C are in arithmetic progression).

for b from 3 to 30 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a+b+c);
end do;
end do;

A336754[n_] := 3*Ceiling[2*Sqrt[n+Round[Sqrt[n]]]]; Array[A336754, 100] (* or *)
Flatten[Array[ConstantArray[3*#, Floor[(#-1)/2]] &, 19, 3]] (* Paolo Xausa, Feb 29 2024 *)

a(n) = A336750(n, 1) + A336750(n, 2) + A336750(n, 3).

a(n) = 3 * A307136(n).

A024164 Number of integer-sided triangles with sides a,b,c, a

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 5, 0, 0, 6, 0, 0, 6, 0, 0, 7, 0, 0, 7, 0, 0, 8, 0, 0, 8, 0, 0, 9, 0, 0, 9, 0, 0, 10, 0, 0, 10, 0, 0, 11, 0, 0, 11, 0, 0, 12, 0, 0, 12, 0, 0, 13, 0, 0, 13, 0, 0, 14, 0, 0, 14, 0, 0, 15, 0, 0, 15, 0, 0, 16, 0, 0, 16

Offset: 1

Clark Kimberling

From Bernard Schott, Oct 10 2020: (Start)
Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.
Equivalently: number of integer-sided triangles such that b = (a+c)/2 with a < c and perimeter n.
All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.
For each perimeter n = 12*k with k>0, there exists one and only one such right integer triangle whose triple is (3k, 4k, 5k).
For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).

Wikipedia, Integer Triangle
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), this sequence (number of triangles whose perimeter = n), A336755 (primitive triples), A336756 (primitive perimeters), A336757 (number of primitive triangles with perimeter = n).

Cf. A005044 (number of integer-sided triangles with perimeter = n).

A024164[n_] := If[Mod[n, 3] == 0, Floor[(n - 3)/6], 0]; Array[A024164, 100] (* Wesley Ivan Hurt, Nov 01 2020 *)
LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{0,0,0,0,0,0,0,0,1},120] (* Harvey P. Dale, Jun 03 2021 *)

If n = 3*k, then a(n) = floor((n-3)/6) = A004526((n-3)/3), otherwise, a(3k+1) = a(3k+2) = 0. - Bernard Schott, Oct 10 2020
From Wesley Ivan Hurt, Nov 01 2020: (Start)
G.f.: x^9/((x^3 - 1)^2*(x^3 + 1)).
a(n) = a(n-3) + a(n-6) - a(n-9).
a(n) = (1 - ceiling(n/3) + floor(n/3)) * floor((n-3)/6). (End)
E.g.f.: (18 + (x - 6)*cosh(x) + (x - 3)*sinh(x) - exp(-x/2)*((9 + 3*exp(x) + x)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Feb 29 2024

A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

9, 16, 25, 36, 49, 64, 49, 64, 81, 100, 121, 81, 121, 100, 121, 169, 196, 121, 169, 144, 169, 196, 225, 256, 289, 169, 289, 361, 196, 225, 256, 289, 324, 361, 400, 441, 225, 289, 361, 256, 289, 361, 484, 529, 289, 361, 441, 529, 625, 324, 361, 400, 441, 484, 529, 576

Offset: 1

Bernard Schott, Jan 05 2021

nonn

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the c's.
For the corresponding primitive triples and miscellaneous properties and references, see A339856.
The terms are all squares >= 9 but they are not in increasing order. For example, a(6) = 64 for triple (25, 40, 64) while a(7) = 49 for triple (36, 42, 49).

			a(1) = 9 for only the smallest such triangle (4, 6, 9) with 6^2 = 4*9, this one corresponds to an obtuse triangle because sqrt(phi) < q = 3/2 < phi, hence C > Pi/2.
a(3) = 25 for only the triple (16, 20, 25) with 16 * 25 = 20^2, this one corresponds to an acute triangle because 1 < q = 5/4 < sqrt(phi), hence C < Pi/2.

Project Euler, Problem 370: Geometric triangles.

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), this sequence (largest side), A339860 (perimeter).
Cf. A336753 (similar for sides in arithmetic progression).
Cf. A335896 (similar for angles in arithmetic progression).

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(c); end if;
end do;
end do;
end do;

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(c, ", "); ); ); ); ); } \\ Michel Marcus, Jan 07 2021

a(n) = A339856(n, 3).

cp's OEIS Frontend

A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A336751 Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A336754 Perimeters in increasing order of integer-sided triangles whose sides a < b < c are in arithmetic progression.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A024164 Number of integer-sided triangles with sides a,b,c, a

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A339859 Largest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula