A335897 -id:A335897 - cp's OEIS frontend

A335893 Primitive triples for integer-sided triangles whose angles A < B < C are in arithmetic progression.

3, 7, 8, 5, 7, 8, 7, 13, 15, 8, 13, 15, 5, 19, 21, 16, 19, 21, 11, 31, 35, 24, 31, 35, 7, 37, 40, 33, 37, 40, 13, 43, 48, 35, 43, 48, 16, 49, 55, 39, 49, 55, 9, 61, 65, 56, 61, 65, 32, 67, 77, 45, 67, 77, 17, 73, 80, 63, 73, 80, 40, 79, 91, 51, 79, 91, 11, 91, 96

Offset: 1

Bernard Schott, Jun 29 2020

The triples are displayed in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side, hence, each triple (a, b, c) is in increasing order.
These three properties below are equivalent:
-> integer-sided triangles whose angles A < B < C are in arithmetic progression,
-> integer-sided triangles such that B = (A+C)/2 with A < C,
-> integer-sided triangles such that A < B < C with B = Pi/3.
When A < B < C are in arithmetic progression with B = A + phi and C = B + phi, then 0 < phi < Pi/3.
The corresponding metric relation between sides is b^2 = a^2 - a*c + c^2.
There exists such primitive triangle iff b^2 is an odd square term of A024612. Hence, the first few middle sides b are 7, 13, 19, 31, 37, 43, 49, 61, 67, ... and b is a term of A004611 \ {1}. Indeed, b cannot be even if the triple is primitive.
As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sines (see link): b/sin(B) = c/sin(C), and c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in the PARI and Maple programs below.
When triple (a, b, c) is solution, then triple (c-a, b, c) is another solution. Hence, for each b odd solution, there exist 2 triples with same middle side b and same largest side c.
The common tangent to the nine-point circle and the incircle of a triangle ABC is parallel to the Euler line iff angles A < B < C are in arithmetic progression (see Crux Mathematicorum for Indian team selection). - Bernard Schott, Apr 14 2022
These triples are called (primitive) Eisenstein triples (Wikipedia). - Bernard Schott, Sep 21 2022

			(3, 7, 8) is a triple for this sequence because from law of cosines (see link), cos(A) = (7^2 + 8^2 - 3^2)/(2*7*8) = 13/14, cos(B) = (8^2 + 3^2 - 7^2)/(2*8*3) = 1/2 and cos(C) = (3^2 + 7^2 - 8^2)/(2*3*7) = -1/7; then, (A+C)/2 = ( arccos(13/14) + arccos(-1/7) )/2 = Pi/3 = B.
Also, arccos(13/14) ~ 21.787 degrees, arccos(1/2) = 60 degrees, arccos(-1/7) ~ 98.213 degrees, so B-A = C-B ~ 38.213 degrees, hence (A, B, C) are in arithmetic progression.
5^2 - 5*8 + 8^2 = 7^2, hence (5, 7, 8) is another triple for triangle whose angles A < B < C are in arithmetic progression.

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

Bill Sands, Indian team selection test 2007, question 5, Crux Mathematicorum, Vol. 36, No. 5 (2010), p. 278.
Eric Weisstein's World of Mathematics, Law of Cosines.
Eric Weisstein's World of Mathematics, Law of Sines.
Wikipedia, Eisenstein triple.
Index to sequences related to Olympiads.

Cf. A004611, A024612.
Cf. A335894 (smallest side), A335895 (middle side), A335896 (largest side), A335897 (perimeter).
Cf. A103606 (primitive Pythagorean triples), A335034 (primitive triples for triangles with two perpendicular medians).

for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a,b,c,c-a,b,c); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print(a, ", ", b, ", ", c, ", "); print(c-a, ", ", b, ", ", c, ", ");););););} \\ Michel Marcus, Jul 15 2020

A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

Original entry on oeis.org

7, 7, 13, 13, 19, 19, 31, 31, 37, 37, 43, 43, 49, 49, 61, 61, 67, 67, 73, 73, 79, 79, 91, 91, 91, 91, 97, 97, 103, 103, 109, 109, 127, 127, 133, 133, 133, 133, 139, 139, 151, 151, 157, 157, 163, 163, 169, 169, 181, 181, 193, 193, 199, 199, 211, 211, 217, 217, 217, 217

Offset: 1

Bernard Schott, Jul 04 2020

nonn

Equivalently, lengths of the middle side b of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to angle B = Pi/3.
Also solutions b of the Diophantine equation b^2 = a^2 - a*c + c^2 with a < b and gcd(a,b) = 1.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple or a solution, then (c-a, b, c) is another solution, so every b in the data is present an even number of times (see examples).
From Bernard Schott, Apr 02 2021: (Start)
Terms are primes of the form 6k+1, or products of primes of the form 6k+1. Three observations:
-> The lengths b are in A004611 \ {1} without repetition, 1 corresponds to the equilateral triangle (1, 1, 1).
-> Every term appears 2^k (k>0) times consecutively and the smallest term that appears 2^k times is precisely A121940(k); see examples.
-> The terms that appear precisely twice consecutively are in A133290. (End)

			b = 7 appears twice because A121940(1) = 7 and:
  7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
  7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
b = 91 appears four times because A121940(2) = 91 and:
  91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
  91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
  91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
  91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
b = 1729 appears eight times because A121940(3) = 1729 and the triples of these 2^3 = 8 triangles are (96, 1729, 1775), (1679, 1729, 1775), (249, 1729, 1840), (1591, 1729, 1840), (656, 1729, 1961), (1305, 1729, 1961), (799, 1729, 1984), (1185, 1729, 1984).

Cf. A004611, A121940, A133290.

Cf. A335893 (triples), A335894 (smallest side), this sequence (middle side), A335896 (largest side), A335897 (perimeter).

for b from 3 to 100 by 2 do
for a from 1 to b-1 do
c := (a+ sqrt(4*b^2-3*a^2))/2;
if igcd(a, b) = 1 and issqr(4*b^2-3*a^2) then print(b); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(a=1, b-1, if (gcd(a, b) == 1, my(d = 4*b^2 - 3*a^2); if (issquare(d), my(c = (a + sqrtint(d))/2); if (denominator(c)==1, print1(b, ", "));););););} \\ Michel Marcus, Jul 05 2020

a(n) = A335893(n, 2).

b is such that b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.

A335894 Smallest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

Original entry on oeis.org

3, 5, 7, 8, 5, 16, 11, 24, 7, 33, 13, 35, 16, 39, 9, 56, 32, 45, 17, 63, 40, 51, 11, 85, 19, 80, 55, 57, 40, 77, 24, 95, 13, 120, 23, 120, 65, 88, 69, 91, 56, 115, 25, 143, 75, 112, 15, 161, 104, 105, 32, 175

Offset: 1

Bernard Schott, Jul 15 2020

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of middle side, and if middle sides coincide, then by increasing order of the largest side, and when largest sides coincide, then by increasing order of the smallest side (see last example). This sequence lists the a's.
Equivalently, lengths of the smallest side a of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to the smallest angle A.
Also, solutions a of the Diophantine equation b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple with a < c/2, then (c-a, b, c) is the following triple because if b^2 = a^2 - a*c + c^2 then also b^2 = (c-a)^2 - (c-a)*c + c^2; hence, for each pair (b,c), there exist two distinct triangles whose smallest sides a_1 and a_2 satisfy a_1 + a_2 = c (see first example).

			For the pair b = 7, c = 8 the two corresponding values of a are 3 and 5 with 3 + 5 = 8 = c because:
    7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
    7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
For b = 91, there exist four corresponding values of a, two for b = 91 and c = 96 that are 11 and 85 with 11 + 85 = 96 = c, and two for b = 91 and c = 99 that are 19 and 80 with 19 + 80 = 99 = c; also these four smallest sides are ordered 11, 85, 19, 80 in the data because:
    91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
    91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
    91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
    91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

Cf. A335893 (triples), A335895 (middle side), A335896 (largest side), A335897 (perimeter).

for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a,c-a); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print1(a, ", ", c-a, ", "); ); ); ); ); } \\ Michel Marcus, Jul 16 2020

a(n) = A335893(n, 1).

a is such that a^2 - c*a + c^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.

A335896 Largest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

Original entry on oeis.org

8, 8, 15, 15, 21, 21, 35, 35, 40, 40, 48, 48, 55, 55, 65, 65, 77, 77, 80, 80, 91, 91, 96, 96, 99, 99, 112, 112, 117, 117, 119, 119, 133, 133, 143, 143, 153, 153, 160, 160, 171, 171, 168, 168, 187, 187, 176, 176, 209, 209, 207, 207, 221, 221, 224, 224, 225, 225

Offset: 1

Bernard Schott, Jul 10 2020

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side. This sequence lists the c's.
Equivalently, lengths of the largest side c of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to the largest angle C.
Also, solutions c of the Diophantine equation b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple, then (c-a, b, c) is another triple, so every c in the data is twice consecutively present according to the corresponding pair (b, c) (see examples).
As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sinus, b/sin(B) = c/sin(C), c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in PARI and Maple programs.
This sequence is not increasing. For example, a(41) = a(42) = 171 for triangle with middle side = 151 while a(43) = a(44) = 168 for triangle with middle side = 157.

			c = 8 appears twice because:
  7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
  7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
c = 96 and c = 99 each appear twice associated with b = 91 because:
  91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
  91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
  91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
  91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

Wikipedia, Law of sines.

Cf. A089025 (terms in increasing order without repetition).

Cf. A335893 (triples), A335894 (smallest side), A335895 (middle side), this sequence (largest side), A335897 (perimeter).

for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(c,c); end if;
end do;
end do;

lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print(c, ", ", c, ", ");););););} \\ Michel Marcus, Jul 15 2020

a(n) = A335893(n, 3).

c satisfies c^2 - a*c + a^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.

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).

A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

15, 28, 45, 66, 91, 120, 40, 153, 190, 231, 276, 84, 325, 378, 435, 496, 144, 77, 561, 630, 703, 104, 780, 220, 861, 946, 1035, 1128, 312, 1225, 170, 1326, 1431, 1540, 126, 420, 209, 1653, 1770, 1891, 2016, 544, 2145, 2278, 299, 2415, 2556, 198, 684, 2701, 350, 2850, 3003, 3160

Offset: 1

Bernard Schott, Apr 15 2021

nonn

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
This sequence is nonincreasing: a(7) = 40 < a(6) = 120.
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
As the metric relation is equivalent to a = m^2 - k^2, b = m*k, c = k^2, with gcd(m,k) = 1 and k < m < 2k, so all terms are of the form m^2 + m*k = m * (m+k) with gcd(m,k) = 1 and k < m < 2k. These perimeters are in increasing order in A106499.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 15 with c < a < b for the first triple (5, 6, 4);
a(7) = 40 with c < b < a for the seventh triple (16, 15, 9);
a(8) = 153 with a < c < b for the eighth triple (17, 72, 64).

Cf. A335897 (similar for A < B < C in arithmetic progression).

Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343066 (side c), A106499 (perimeters in increasing order).

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and igcd(a, sqrt(d), c)=1 and abs(a-c)

lista(nn) = {for (a = 2, nn, for (c = 3, a^2\2, my(d = c*(a+c)); if (issquare(d) && (gcd([a, sqrtint(d), c])==1) && (abs(a-c)Michel Marcus, May 12 2022

a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).

a(n) = A343064(n) + A343065(n) + A343066(n).

A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.

Original entry on oeis.org

19, 37, 61, 91, 109, 129, 127, 169, 193, 219, 247, 217, 273, 271, 301, 367, 403, 331, 399, 397, 433, 471, 511, 553, 597, 469, 637, 733, 547, 589, 633, 679, 727, 777, 829, 883, 631, 723, 823, 721, 769, 871, 1039, 1099, 817, 921, 1033, 1153, 1281, 919, 973, 1029, 1087

Offset: 1

Bernard Schott, Jan 08 2021

nonn

The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order.
These perimeters are of the form r^2 + r*s + s^2, r < s, gcd(r, s) = 1 and q = r/s (A034017), so they are all odd but not in increasing order. For example, a(6) = 129 for triple (25, 40, 64) while a(7) = 127 for triple (36, 42, 49).
For the corresponding primitive triples and miscellaneous properties, see A339859.

			a(1) = 19 = 4+6+9 for the smallest such triangle (4, 6, 9) with 4 * 9 = 6^2 and a ratio q = 3/2.
a(2) = 37 = 9+12+16 for the triple (9, 12, 16) with 9 * 16 = 12^2 and a ratio q = 4/3.

Project Euler, Problem 370: Geometric triangles.

Cf. A339856 (triples), A339857 (smallest side), A339858 (middle side), A339859 (largest side), this sequence (perimeter).
Cf. A336754 (similar for sides in arithmetic progression).
Cf. A335897 (similar for angles in arithmetic progression).
Subsequence of A034017.

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(a+b+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(a+b+c, ", ")); ); ); ); } \\ Michel Marcus, Jan 08 2021

a(n) = A339856(n, 1) + A339856(n, 2) + A339856(n, 3).

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A335893 Primitive triples for integer-sided triangles whose angles A < B < C are in arithmetic progression.

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A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

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A335894 Smallest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

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A335896 Largest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

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

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A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

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A339860 Perimeter of primitive integer-sided triangles whose sides a < b < c form a geometric progression.

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