A336757 -id:A336757 - 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).

A336755 Primitive triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 3, 5, 7, 4, 5, 6, 5, 6, 7, 4, 7, 10, 5, 7, 9, 6, 7, 8, 5, 8, 11, 7, 8, 9, 5, 9, 13, 7, 9, 11, 8, 9, 10, 7, 10, 13, 9, 10, 11, 6, 11, 16, 7, 11, 15, 8, 11, 14, 9, 11, 13, 10, 11, 12, 7, 12, 17, 11, 12, 13, 7, 13, 19, 8, 13, 18, 9, 13, 17, 10, 13, 16, 11, 13, 15, 12, 13, 14

Offset: 1

Bernard Schott, Sep 07 2020

The triples are displayed in increasing order of perimeter (equivalently in increasing order of middle side) and if perimeters coincide then by increasing order of the smallest side; also, each triple (a, b, c) is in increasing order.
When b is prime, all the corresponding triples in A336750 are primitive triples.
The only right integer triangle in the data corresponds to the triple (3, 4, 5).
The number of primitive such triangles whose middle side = b is equal to A023022(b) for b >= 3.
For all the triples (primitive or not), miscellaneous properties and references, see A336750.

			The table begins:
  2, 3, 4;
  3, 4, 5;
  3, 5, 7;
  4, 5, 6;
  5, 6, 7;
  4, 7, 10;
  5, 7, 9;
  6, 7, 8;
The smallest such primitive triple 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 exists only one primitive triangle with perimeter = 18 whose triple is (5, 6, 7), because (4, 6, 8) is not a primitive triple.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A336750 (triples, primitive or not), this sequence (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).
Cf. A103606 (similar for primitive Pythagorean triples).
Cf. A023022.

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

Select[Flatten[Table[{a, b, 2*b-a}, {b, 3, 20}, {a, b-Floor[(b-1)/2], b-1}], 1], GCD @@ # == 1 &] (* Paolo Xausa, Feb 28 2024 *)

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

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

A336756 Perimeters in increasing order of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

9, 12, 15, 15, 18, 21, 21, 21, 24, 24, 27, 27, 27, 30, 30, 33, 33, 33, 33, 33, 36, 36, 39, 39, 39, 39, 39, 39, 42, 42, 42, 45, 45, 45, 45, 48, 48, 48, 48, 51, 51, 51, 51, 51, 51, 51, 51, 54, 54, 54, 57, 57, 57, 57, 57, 57, 57, 57, 57, 60, 60, 60, 60, 63, 63, 63, 63, 63, 63

Offset: 1

Bernard Schott, Sep 16 2020

nonn

Equivalently: perimeters of primitive integer-sided triangles such that b = (a+c)/2 with a < c.
As perimeter = 3 * middle side, these perimeters p are all multiples of 3 and each term p appears consecutively A023022(p/3) = phi(p/3)/2 times for p >= 9.
Remark, when the middle side is prime, then the number of primitive triangles with a perimeter p = 3*b equals phi(p/3)/2 = (b-1)/2 = (p-3)/6 and in this case, all the triangles are primitive (see A336754).
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 the Pythagorean triple (3, 4, 5).
Perimeter = 15 for the two triples (3, 5, 7) and (4, 5, 6).
There only exists one primitive triangle with perimeter = 18 whose triple is (5, 6, 7), because (4, 6, 8) is not a primitive triple.

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

Cf. A336754 (perimeters, primitive or not), A336755 (primitive triples), this sequence (perimeters of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).

Cf. A023022.

for b from 3 to 21 do
for a from b-floor((b-1)/2) to b -1 do
c := 2*b - a;
if gcd(a,b)=1 and gcd(b,c)=1 then print(a+b+c); end if;
end do;
end do;

Flatten[Array[ConstantArray[3*#, EulerPhi[#]/2] &, 20, 3]] (* Paolo Xausa, Feb 29 2024 *)

lista(nn) = {my(list=List()); for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); if (gcd([a, b, c]) == 1, listput(list, a+b+c);););); Vec(list);} \\ Michel Marcus, Sep 16 2020

A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

6, 84, 126, 156, 210, 456, 546, 570, 1116, 1170, 1176, 1554, 2046, 2220, 2394, 3096, 3216, 3294, 3354, 3924, 4740, 5124, 6006, 6180, 6510, 7326, 7446, 8760, 9030, 9264, 9906, 10374, 10920, 11466, 12684, 13104, 15210, 16170, 16296, 16716, 17556, 18060, 18090, 18354, 22134, 22860, 23550

Offset: 1

Bernard Schott, Feb 04 2022

nonn

Middle side b is necessarily even, and the two other sides are odd, so all the areas are even numbers.

If b is the middle side with b even >= 4, if k odd = b-a = c-b with 1 <= k <= b/2 - 1, if gcd(b,k) = 1, then, we have area S = sqrt(3*b^2*(b^2-4*k^2))/4.

			a(1) = 6 corresponds to the Pythagorean triple (3, 4, 5), this is the unique right integer-sided triangle in this sequence.
a(2) = 84 for triple (13, 14, 15) (see MacNeill link).
a(3) = 126 for triple (15, 28, 41) (see Penguin reference, entry 126).
a(4) = 156 for triple (15, 26, 37) (see MacNeill link).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 126, page 122.

John MacNeill, 13, 14, 15 and 15, 26, 37, Mathematical Spectrum, Vol. 21, No. 3, 1989, pp. 83-84.

Cf. A336750, A336755, A336750, A336756, A336757.

Subsequence of A188158.

lista(nn) = {my(list = List()); for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); if (gcd([a,b,c]) == 1, my(p = (a+b+c)/2); if (issquare(x=p*(p-a)*(p-b)*(p-c)), listput(list, sqrtint(x)));););); vecsort(Vec(list));} \\ Michel Marcus, Feb 05 2022

Missing terms added by Michel Marcus, Feb 05 2022

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

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A336755 Primitive triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

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

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A336756 Perimeters in increasing order of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

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A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

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