A024164 -id:A024164 - 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).

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).

A336757 Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 0, 2, 0, 0, 6, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 8, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 11, 0, 0, 4

Offset: 1

Bernard Schott, Sep 20 2020

nonn

Equivalently: number of primitive integer-sided triangles such that b = (a+c)/2 with a < c and perimeter = n.
As the perimeter of these triangles = 3*b where b is the middle side, a(n) >= 1 iff n = 3*b, with b >= 3.
When b is prime, all the triangles of perimeter n = 3*b are primitive, hence in this case: a(n) = A024164(n).
For the corresponding triples (primitive or not), miscellaneous properties and references, see A336750.

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for the Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
a(18) = 1 for the triple (5, 6, 7); the other triple (4, 6, 8) corresponding to a perimeter = 18 is not a primitive triple.

Cf. A336750 (triples, primitive or not), A336755 (primitive triples), A336756 (perimeters of primitive triangles).
Cf. A024164 (number of such triangles, primitive or not).
Similar sequences: A005044 (integer-sided triangles), A024155 (right triangles), A070201 (with integral inradius).
Cf. A000010, A023022.

For n = 3*b, b >= 3, a(n) = A023022(b) = A000010(b)/2, otherwise a(n) = 0.

A024165 Number of integer-sided triangles with sides a,b,c, a b - a.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 4, 2, 2, 4, 4, 2, 6, 4, 4, 6, 6, 4, 9, 6, 6, 9, 9, 6, 12, 9, 9, 12, 12, 9, 16, 12, 12, 16, 16, 12, 20, 16, 16, 20, 20, 16, 25, 20, 20, 25, 25, 20, 30, 25, 25, 30, 30, 25, 36, 30, 30, 36, 36, 30, 42, 36, 36, 42, 42, 36, 49, 42, 42, 49, 49

Offset: 1

Clark Kimberling

Same as A025828 with zeros prepended. - Joerg Arndt, Nov 04 2014

G. C. Greubel, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,1,-1,0,-1,-1,0,0,1).

Cf. A024163, A024164, A025828.

R:=PowerSeriesRing(Rationals(), 100);
[0,0,0,0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^13/((1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Jul 03 2021

LinearRecurrence[{0,0,1,1,0,1,-1,0,-1,-1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1},100] (* Harvey P. Dale, Sep 04 2017 *)

a(n) = ((n-1)\3 - n\4)*((n-1)\3 + n\4 - n\2) \\ Hoang Xuan Thanh, Aug 31 2025

def A024165_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( x^13/((1-x^3)*(1-x^4)*(1-x^6)) ).list()
a=A024165_list(100); a[1:] # G. C. Greubel, Jul 03 2021

G.f.: x^13/((1-x^3)*(1-x^4)*(1-x^6)). - Tani Akinari, Nov 04 2014
From Robert Israel, Nov 04 2014: (Start)
a(n) = a(n-3) + a(n-4) + a(n-6) - a(n-7) - a(n-9) - a(n-10) + a(n-13) for n >= 14.
a(6*n) = (2*n^2 - 8*n + 7 + (-1)^n)/8, n >= 1.
a(6*n+1) = a(6*n+4) = a(6*n+5) = (2*n^2 - 1 + (-1)^n)/8.
a(6*n+2) = a(6*n+3) = (2*n^2 - 4*n + 1 - (-1)^n)/8. (End)
From Hoang Xuan Thanh, Aug 31 2025: (Start)
a(n) = floor((n^2 -5*n +40 -(n-13)*(3*(-1)^n +8*((n+2) mod 3)) -12*((n+5) mod 6))/144).
a(n) = (floor((n-1)/3) - floor(n/4))*(floor((n-1)/3) + floor(n/4) - floor(n/2)). (End)

A117910 Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 2, 3, 5, 3, 3, 6, 3, 4, 7, 4, 4, 8, 4, 5, 9, 5, 5, 10, 5, 6, 11, 6, 6, 12, 6, 7, 13, 7, 7, 14, 7, 8, 15, 8, 8, 16, 8, 9, 17, 9, 9, 18, 9, 10, 19, 10, 10, 20, 10, 11, 21, 11, 11, 22, 11, 12, 23, 12, 12, 24, 12, 13, 25, 13, 13, 26, 13, 14, 27, 14

Offset: 0

Paul Barry, Apr 01 2006

Diagonal sums of A117908.

Appears to be a permutation of floor((n+5)/5).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Cf. A002264, A014125, A024164, A144677, A117908, A152467, A175676.

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) )); // G. C. Greubel, Oct 21 2021

CoefficientList[Series[(1+x+x^2+x^4)/((1-x^3)(1-x^6)),{x,0,100}],x] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{1,1,1,1,2,1,2,3,2},100] (* Harvey P. Dale, Apr 10 2014 *)
Table[If[Mod[n,3]==1, Mod[Binomial[n+2,3], n+2], Floor[(n+6)/6]], {n, 0, 100}] (* G. C. Greubel, Nov 18 2021 *)

def A117910_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) ).list()
A117910_list(100) # G. C. Greubel, Oct 21 2021

a(n) = a(n-3) + a(n-6) - a(n-9).
a(n) = Sum_{k=0..floor(n/2)} 0^abs(L(C(n-k,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.
From G. C. Greubel, Nov 18 2021: (Start)
a(n) = A152467(n+3) + A152467(n+6) if n == 1 (mod 3), otherwise A152467(n+6).
a(n) = A175676(n+2) if n == 1 (mod 3), otherwise A152467(n+6).
a(n) = A002264(n+3) if n == 1 (mod 3), otherwise A152467(n+6). (End)

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

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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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A336757 Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.

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A024165 Number of integer-sided triangles with sides a,b,c, a b - a.

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A117910 Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).

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