cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 33, 35, 37, 39, 39, 40, 41, 43, 45, 47, 48, 49, 51, 51, 53, 55, 56, 56, 57, 57, 59, 61, 63, 64, 65, 67, 69, 69, 71, 72, 72, 73, 75, 75, 77, 79, 80, 81, 83, 85, 85, 87, 87, 88, 88, 89, 91, 93, 93, 95, 95, 96, 97, 99
Offset: 1

Views

Author

Bernard Schott, Apr 10 2021

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
Equivalently, length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
   c < a < b for the smallest side a = 5 and triple (5, 6, 4).
The first side a for which there exist two distinct triangles with B = 2C is for a = 33 with the two other types of examples,
   c < b < a with triple (33, 28, 16),
   a < c < b with triple (33, 272, 256).

Links

Crossrefs

Cf. A353619 (similar, but with B = 3*C).
Cf. A343063 (triples), this sequence (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).
Cf. A106505 (sides a without repetition), A106506 (sides a sorted on perimeter).

Programs

  • Maple
    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)

Formula

a(n) = A343063(n, 1).

A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

6, 12, 20, 30, 42, 56, 15, 72, 90, 110, 132, 35, 156, 182, 210, 240, 63, 28, 272, 306, 342, 40, 380, 99, 420, 462, 506, 552, 143, 600, 70, 650, 702, 756, 45, 195, 88, 812, 870, 930, 992, 255, 1056, 1122, 130, 1190, 1260, 77, 323, 1332, 154, 1406, 1482, 1560, 399, 1640, 1722, 66, 1806, 208, 1892, 117, 483, 1980, 2070, 238
Offset: 1

Views

Author

Bernard Schott, Apr 11 2021

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
This sequence is not increasing because a(7) = 15 < a(6) = 56 (A106430).
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
Equivalently, length of side opposite to the greater of the two angles, one being the double of the other.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
c < a < b for the first triple (5, 6, 4) with b = 6.
c < b < a for the second triple (16, 15, 9) with b = 15.
a < c < b for the seventh triple (7, 12, 9) with b = 12.

Links

Crossrefs

Cf. A335895 (similar for A < B < C in arithmetic progression).
Cf. A343063 (triples), A343064 (side a), A343066 (side c), A343067 (perimeter).
Cf. A106420 (sides b sorted on perimeter), A106430 (sides b in increasing order).

Programs

  • Maple
    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)

Formula

a(n) = A343063(n, 2).

A343066 Side c of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 9, 64, 81, 100, 121, 25, 144, 169, 196, 225, 49, 16, 256, 289, 324, 25, 361, 81, 400, 441, 484, 529, 121, 576, 49, 625, 676, 729, 25, 169, 64, 784, 841, 900, 961, 225, 1024, 1089, 100, 1156, 1225, 49, 289, 1296, 121, 1369, 1444, 1521, 361, 1600, 1681, 36, 1764, 169, 1849, 81, 441, 1936, 2025, 196
Offset: 1

Views

Author

Bernard Schott, Apr 12 2021

Keywords

Comments

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
This sequence is not increasing because a(7) = 9 < a(6) = 49.
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
All terms are perfect squares >= 4.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
c = 4 with c < a < b for the first triple (5, 6, 4).
c = 9 with c < b < a for the seventh triple (16, 15, 9).
c = 16 with a < c < b for the third triple (9, 20, 16).

Links

Crossrefs

Cf. A335896 (similar for A < B < C in arithmetic progression).
Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343067 (perimeter).

Programs

  • Maple
    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)

Formula

a(n) = A343063(n, 3)

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

Views

Author

Bernard Schott, Apr 15 2021

Keywords

Comments

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.

Examples

			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).

Crossrefs

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).

Programs

  • Maple
    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)
  • PARI
    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

Formula

a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).
a(n) = A343064(n) + A343065(n) + A343066(n).

A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

The terms are proposed without repetition. For example, there exist two such triangles with a length of side = 33. They correspond respectively to s^2 - r^2 = 33 (see formula) with (r, s) = (4, 7) and sides (33, 28, 16), and the other triangle with (r, s) = (16, 17) and sides (33, 272, 256). Lengths = 39, 51, 57, 69, 75, ... correspond to two distinct triangles ... The lengths of these sides are proposed with repetition in A343064. - Bernard Schott, Apr 22 2021

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430, A321499.
Cf. A343063, A343064, A343065, A343066, A343067.

Formula

Values s^2 - r^2, where r
Conjecture: for n>2, a(n+5) = a(n) + 8. - Ralf Stephan, Nov 16 2010.
Empirical g.f.: -x*(x^7+x^6+3*x^5-2*x^4-2*x^3-2*x^2-2*x-5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Oct 05 2013

Extensions

Extended by Ray Chandler, May 09 2005

A353618 Three-column array giving list of primitive triples for integer-sided triangles whose angle B = 3*C.

Original entry on oeis.org

3, 10, 8, 35, 48, 27, 119, 132, 64, 112, 195, 125, 279, 280, 125, 20, 357, 343, 253, 504, 343, 539, 510, 216, 552, 665, 343, 91, 792, 729, 923, 840, 343, 533, 840, 512, 476, 1035, 729, 1455, 1288, 512, 224, 1485, 1331, 1504, 1575, 729, 17, 1740, 1728, 799
Offset: 1

Views

Author

Bernard Schott, Apr 30 2022

Keywords

Comments

This sequence is inspired by the 1st problem proposed during the 46th Czech and Slovak Mathematical Olympiad 1997 (see link).
The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c.
If in triangle ABC, B = 3*C, then the corresponding metric relations between sides are c*a^2= (b-c)^2 * (b+c). <===> a/(b-c) = sqrt(1+b/c).
This metric relation is equivalent to a = m(m^2-2k^2), b = k(m^2-k^2), c = k^3, gcd(k,m) = 1 and sqrt(2) * k < m < 2*k; hence every c is a cube number and always c < b.
When A <> 3*Pi/7 and A <> Pi/5, table below shows there exist these 3 possible configurations: c < b < a; c < a < b and a < c < b:
----------------------------------------------------------------------------
| A | Pi | decr. | 3*Pi/7 | decr. | Pi/5 | decr. | 0 |
---------------------------------------------------------------------------
| B | 0 | incr. | 3*Pi/7 | incr. | 3*Pi/5 | incr. | 3*Pi/4 |
----------------------------------------------------------------------------
| C | 0 | incr. | Pi/7 | incr. | Pi/5 | incr. | Pi/4 |
----------------------------------------------------------------------------
| < | No | c < b < a | c < b=a | c < a < b | c=a < b | a < c < b | No |
----------------------------------------------------------------------------
where 'No' means there is no such corresponding triangle.
If (A,B,C) = (3*Pi/7,3*Pi/7,Pi/7) then a = b with c = 2*a*cos(Pi/7), so isosceles ABC is not an integer-sided triangle.
If (A,B,C) = (Pi/5,3*Pi/5,Pi/5) then a = c with b = a*(1+sqrt(5))/2, so ABC is not an integer-sided triangle.

Examples

			The table begins:
    3,  10,   8;
   35,  48,  27;
  119, 132,  64;
  112, 195, 125;
  279, 280, 125;
   20, 357, 343;
  253, 504, 343,
  539, 510, 216;
................
The smallest such triangle is (3,10,8), it is of type a < c < b with 3/(10-8) = sqrt(1+10/8) = 3/2.
The 2nd triple (35, 48, 27) is of type c < a < b with 35/(48-27) = sqrt(1+48/27) = 5/3.
The 8th triple (539, 510, 216) is the first of type c < b < a with  539/(510-216) = sqrt(1+510/216) = 11/6.

Links

Crossrefs

Cf. A335893 (A < B < C are in arithmetic progression), A343063 (B = 2*C).

Programs

  • Maple
    for b from 1 to 2500 do
for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(a,b,q^3); end if;
end do;
end do;
  • PARI
    lista(nn) = {for (b = 1, nn, for (q = 2, sqrtnint(b-1, 3), if (issquare(z=1+b/q^3), a = (b-q^3) * sqrtint(numerator(z))/sqrtint(denominator(z)); if ((q^3 < b) && (gcd([a, b, q]) == 1) && ((b-q^3) < a) && (a < b+q^3), print1([a, b, q^3], ", ")););););} \\ Michel Marcus, May 11 2022
  • Python
    from math import gcd
from itertools import count, islice
from sympy import integer_nthroot
def A353618_gen(): # generator of terms
    for b in count(1):
        q, c = 2, 8
        while c < b:
            d = (b-c)**2*(b+c)
            s, t = divmod(d,c)
            if t == 0:
                a, r = integer_nthroot(s,2)
                if r and b-c < a < b+c and gcd(a,b,q) == 1:
                    yield from (a, b, c)
            c += q*(3*q+3)+1
            q += 1
A353618_list = list(islice(A353618_gen(),30)) # Chai Wah Wu, May 14 2022
Showing 1-6 of 6 results.

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