A321770 -id:A321770 - cp's OEIS frontend

A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).

2, 3, 5, 4, 4, 8, 7, 8, 12, 9, 7, 9, 6, 5, 11, 10, 13, 19, 14, 12, 16, 11, 11, 21, 18, 19, 29, 22, 16, 20, 13, 10, 18, 15, 14, 22, 17, 11, 13, 8, 6, 14, 13, 18, 26, 19, 17, 23, 16, 18, 34, 29, 30, 46, 35, 25, 31, 20, 17, 31, 26, 25, 39, 30, 20, 24, 15, 14, 30

Offset: 1

Rémy Sigrist, Nov 18 2018

This sequence and A321783 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

This sequence is "i" from the construction in A321768. It takes ternary digits of k-1 from most to least significant. Here the result is the same going instead least to most, due to how the relevant matrix product is related to its reversal. As a flat sequence this means a(A351702(n)) = a(n) unchanged. - Kevin Ryde, Mar 10 2022

			The first rows are:
   2
   3, 5, 4
   4, 8, 7, 8, 12, 9, 7, 9, 6

Rémy Sigrist, Rows n = 1..9, flattened
Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree, "row-wise p".
Robert Saunders and Trevor Randall, The Family Tree of the Pythagorean Triplets Revisited, Mathematical Gazette, item 78.12, volume 78, July 1994, pages 190-193, see page 192 tree terms "m" by columns.
Index entries related to Pythagorean Triples

Cf. A000129, A321768, A321770, A321783.
Cf. A001542 (row sums).
Cf. A351702 (product reversal permutation).

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[1, 1] + t[3, 1])/2))

Empirically:
- T(n, 1) = n + 1,
- T(n, (3^(n-1) + 1)/2) = A000129(n + 1),
- T(n, 3^(n-1)) = 2 * n.

A321783 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321770(n, k) - A321768(n, k))/2).

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 2, 5, 5, 2, 4, 4, 1, 4, 4, 3, 8, 8, 3, 7, 7, 2, 8, 8, 5, 12, 12, 5, 9, 9, 2, 7, 7, 4, 9, 9, 4, 6, 6, 1, 5, 5, 4, 11, 11, 4, 10, 10, 3, 13, 13, 8, 19, 19, 8, 14, 14, 3, 12, 12, 7, 16, 16, 7, 11, 11, 2, 11, 11, 8, 21, 21, 8, 18, 18, 5, 19, 19

Offset: 1

Rémy Sigrist, Nov 19 2018

This sequence and A321782 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

			The first rows are:
   1
   2, 2, 1
   3, 3, 2, 5, 5, 2, 4, 4, 1

Rémy Sigrist, Rows n = 1..9, flattened
Kevin Ryde, Trees of Primitive Pythagorean Triples, see section UAD Tree, "row-wise q".
Robert Saunders and Trevor Randall, The Family Tree of the Pythagorean Triplets Revisited, Mathematical Gazette, item 78.12, volume 78, July 1994, pages 190-193, see page 192 tree terms "n" by columns.
Index entries related to Pythagorean Triples

Cf. A000129, A321768, A321770, A321782.

Cf. A001653 (row sums).

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[1, 1])/2))

Empirically:
- T(n, 1) = n,
- T(n, (3^(n-1) + 1)/2) = A000129(n),
- T(n, 3^(n-1)) = 1.

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

Rémy Sigrist, Nov 22 2018

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

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

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

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

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]))

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.

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

Rémy Sigrist, Nov 30 2018

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.

			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.

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

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

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])

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 At, Bt, 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

Rémy Sigrist, Nov 18 2018

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)).

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

Rémy Sigrist, Rows n = 1..9, flattened
Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree "row-wise A leg".
Wikipedia, Tree of primitive Pythagorean triples
Index entries related to Pythagorean Triples

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.

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])

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.

A321769 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 At, Bt, 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 second component of P(n, k).

Original entry on oeis.org

4, 12, 20, 8, 24, 48, 28, 80, 120, 36, 56, 72, 12, 40, 88, 60, 208, 304, 84, 168, 224, 44, 176, 336, 180, 456, 696, 220, 288, 360, 52, 140, 252, 120, 252, 396, 136, 132, 156, 16, 60, 140, 104, 396, 572, 152, 340, 460, 96, 468, 884, 464, 1140, 1748, 560, 700

Offset: 1

Rémy Sigrist, Nov 18 2018

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

			The first rows are:
   4
   12, 20, 8
   24, 48, 28, 80, 120, 36, 56, 72, 12

Rémy Sigrist, Rows n = 1..9, flattened
Wikipedia, Tree of primitive Pythagorean triples
Index entries related to Pythagorean Triples

See A321768 and A321770 for the other components.

Cf. A046092, A046729.

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[2, 1])

Empirically:
- T(n, 1) = A046092(n),
- T(n, (3^(n-1) + 1)/2) = A046729(n),
- T(n, 3^(n-1)) = 4 * n.

cp's OEIS Frontend

A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).

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A321783 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321770(n, k) - A321768(n, k))/2).

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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)).

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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).

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