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.

A321784 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt(A321769(n, k) + A321770(n, k)).

Original entry on oeis.org

3, 5, 7, 5, 7, 11, 9, 13, 17, 11, 11, 13, 7, 9, 15, 13, 21, 27, 17, 19, 23, 13, 19, 29, 23, 31, 41, 27, 25, 29, 15, 17, 25, 19, 23, 31, 21, 17, 19, 9, 11, 19, 17, 29, 37, 23, 27, 33, 19, 31, 47, 37, 49, 65, 43, 39, 45, 23, 29, 43, 33, 41, 55, 37, 31, 35, 17
Offset: 1

Views

Author

Rémy Sigrist, Nov 22 2018

Keywords

Comments

This sequence and A321785 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

Examples

			The first rows are:
   3
   5, 7, 5
   7, 11, 9, 13, 17, 11, 11, 13, 7

Links

Crossrefs

Cf. A001333, A321768, A321769, A321770, A321785.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (sqrtint(t[2, 1] + t[3, 1]))

Formula

Empirically:
- T(n, 1) = 2*n + 1,
- T(n, (3^(n-1) + 1)/2) = A001333(n+1),
- T(n, 3^(n-1)) = 2*n + 1.

A321785 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt(A321769(n, k)^2 - A321768(n, k)^2).

Original entry on oeis.org

1, 1, 3, 3, 1, 5, 5, 3, 7, 7, 3, 5, 5, 1, 7, 7, 5, 11, 11, 5, 9, 9, 3, 13, 13, 7, 17, 17, 7, 11, 11, 3, 11, 11, 5, 13, 13, 5, 7, 7, 1, 9, 9, 7, 15, 15, 7, 13, 13, 5, 21, 21, 11, 27, 27, 11, 17, 17, 5, 19, 19, 9, 23, 23, 9, 13, 13, 3, 19, 19, 13, 29, 29, 13, 23
Offset: 1

Views

Author

Rémy Sigrist, Nov 22 2018

Keywords

Comments

This sequence and A321784 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

Examples

			The first rows are:
   1
   1, 3, 3
   1, 5, 5, 3, 7, 7, 3, 5, 5

Links

Crossrefs

Cf. A001333, A321768, A321769, A321784.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (sqrtint(t[3, 1] - t[2, 1]))

Formula

Empirically:
- T(n, 1) = 1,
- T(n, (3^(n-1) + 1)/2) = A001333(n),
- T(n, 3^(n-1)) = 2*n - 1.

A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

Original entry on oeis.org

6, 30, 210, 60, 84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210, 180, 4620, 2730, 10920, 45144, 7854, 7980, 23184, 2574, 5016, 63336, 26910, 49476, 242556, 50490, 25200, 57420, 4290, 3570, 34650, 12540, 14490, 79794, 18564, 5610, 10374, 504, 330, 11970, 7956
Offset: 1

Views

Author

Rémy Sigrist, Nov 29 2018

Keywords

Comments

This sequence gives the areas of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.
If we order the terms in this sequence and keep duplicates then we obtain A024406.

Examples

			The first rows are:
   6
   30, 210, 60
   84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210
T(1,1) corresponds to the area of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 * 4 / 2 = 6.

Links

Crossrefs

Cf. A024406, A029549, A055112, A069072, A321768, A321769.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] * t[2, 1] / 2)

Formula

Empirically:
- T(n, 1) = A055112(n),
- T(n, (3^(n-1) + 1)/2) = A029549(n),
- T(n, 3^(n-1)) = A069072(n-1).

A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

Original entry on oeis.org

12, 30, 70, 40, 56, 176, 126, 208, 408, 198, 154, 234, 84, 90, 330, 260, 546, 1026, 476, 456, 736, 286, 418, 1218, 828, 1178, 2378, 1188, 800, 1160, 390, 340, 900, 570, 644, 1364, 714, 374, 494, 144, 132, 532, 442, 1044, 1924, 874, 918, 1518, 608, 1116, 3196
Offset: 1

Views

Author

Rémy Sigrist, Nov 30 2018

Keywords

Comments

This sequence gives the perimeters of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.
If we order the terms in this sequence and keep duplicates then we obtain A024364.

Examples

			The first rows are:
   12
   30, 70, 40
   56, 176, 126, 208, 408, 198, 154, 234, 84
T(1,1) corresponds to the perimeter of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 + 4 + 5 = 12.

Links

Crossrefs

Cf. A001542, A002939, A024364, A033586, A321768, A321769, A321770.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] + t[2, 1] + t[3, 1])

Formula

Empirically:
- T(n, 1) = A002939(n+1),
- T(n, (3^(n-1) + 1)/2) = A001542(n+1),
- T(n, 3^(n-1)) = A033586(n).

A321768 Consider the ternary tree of triples P(n, k) with n > 0 and 0 < k <= 3^(n-1), such that P(1, 1) = [3; 4; 5] and each triple t on some row branches to the triples A*t, B*t, C*t on the next row (with A = [1, -2, 2; 2, -1, 2; 2, -2, 3], B = [1, 2, 2; 2, 1, 2; 2, 2, 3] and C = [-1, 2, 2; -2, 1, 2; -2, 2, 3]); T(n, k) is the first component of P(n, k).

Original entry on oeis.org

3, 5, 21, 15, 7, 55, 45, 39, 119, 77, 33, 65, 35, 9, 105, 91, 105, 297, 187, 95, 207, 117, 57, 377, 299, 217, 697, 459, 175, 319, 165, 51, 275, 209, 115, 403, 273, 85, 133, 63, 11, 171, 153, 203, 555, 345, 189, 429, 247, 155, 987, 777, 539, 1755, 1161, 429
Offset: 1

Views

Author

Rémy Sigrist, Nov 18 2018

Keywords

Comments

The tree P runs uniquely through every primitive Pythagorean triple.
The ternary tree is built as follows:
- for any n and k such that n > 0 and 0 < k <= 3^(n-1):
- P(n, k) is a column vector,
- P(n+1, 3*k-2) = A * P(n, k),
- P(n+1, 3*k-1) = B * P(n, k),
- P(n+1, 3*k) = C * P(n, k).
All terms are odd.
Every primitive Pythagorean triple (a, b, c) can be characterized by a pair of parameters (i, j) such that:
- i > j > 0 and gcd(i, j) = 1 and i and j are of opposite parity,
- a = i^2 - j^2,
- b = 2 * i * j,
- c = i^2 + j^2,
- A321782(n, k) and A321783(n, k) respectively give the value of i and of j pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).
Every primitive Pythagorean triple (a, b, c) can also be characterized by a pair of parameters (u, v) such that:
- u > v > 0 and gcd(u, v) = 1 and u and v are odd,
- a = u * v,
- b = (u^2 - v^2) / 2,
- c = (u^2 + v^2) / 2,
- A321784(n, k) and A321785(n, k) respectively give the value of u and of v pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).

Examples

			The first rows are:
   3
   5, 21, 15
   7, 55, 45, 39, 119, 77, 33, 65, 35

Links

Crossrefs

See A321769 and A321770 for the other components.
See A322170 for the corresponding areas.
See A322181 for the corresponding perimeters.
Cf. A321782, A321783, A321784, A321785.
Cf. A046727.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n,k) = my (t=[3;4;5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1,1])

Formula

Empirically:
- T(n, 1) = 2*n + 1,
- T(n, (3^(n-1) + 1)/2) = A046727(n),
- T(n, 3^(n-1)) = 4*n^2 - 1.

A321770 Consider the tree of triples P(n, k) with n > 0 and 0 < k <= 3^(n-1), such that P(1, 1) = [3; 4; 5] and each triple t on some row branches to the triples A*t, B*t, C*t on the next row (with A = [1, -2, 2; 2, -1, 2; 2, -2, 3], B = [1, 2, 2; 2, 1, 2; 2, 2, 3] and C = [-1, 2, 2; -2, 1, 2; -2, 2, 3]); T(n, k) is the third component of P(n, k).

Original entry on oeis.org

5, 13, 29, 17, 25, 73, 53, 89, 169, 85, 65, 97, 37, 41, 137, 109, 233, 425, 205, 193, 305, 125, 185, 505, 349, 505, 985, 509, 337, 481, 173, 149, 373, 241, 277, 565, 305, 157, 205, 65, 61, 221, 185, 445, 797, 377, 389, 629, 265, 493, 1325, 905, 1261, 2477
Offset: 1

Views

Author

Rémy Sigrist, Nov 18 2018

Keywords

Comments

The tree P runs uniquely through every primitive Pythagorean triple.
See A321768 for additional comments about P.
All terms are odd.

Examples

			The first rows are:
   5
   13, 29, 17
   25, 73, 53, 89, 169, 85, 65, 97, 37

Links

Crossrefs

See A321768 and A321769 for the other components.
Cf. A001653, A001844, A053755.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[3, 1])

Formula

a(n)^2 = A321768(n)^2 + A321769(n)^2.
Empirically:
- T(n, 1) = A001844(n),
- T(n, (3^(n-1) + 1)/2) = A001653(n+1),
- T(n, 3^(n-1)) = A053755(n).
Showing 1-6 of 6 results.

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