A103605 -id:A103605 - cp's OEIS frontend

A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

1, 2, 8, 27, 39, 54, 73, 98, 133, 186, 273, 426, 709, 1250, 2305, 4386, 8517, 16746, 33169, 65978, 131557, 262674, 524865, 1049202, 2097829, 4195034, 8389393, 16778058, 33555333, 67109826, 134218753, 268436546, 536872069, 1073743050, 2147484945, 4294968666, 8589936037, 17179870706

Offset: 0

Scott R. Shannon, Sep 08 2021

nonn

This sequence uses the same rules as A347594 except here all numbers must be unique. Up to 10^5 terms all terms are larger than the previous term; it is unknown if this holds for all terms as n->infinity.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 8 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 8 = 16 = 4^2. Note that although 8 + 1 = 9 = 3^2, 1 cannot be chosen as a(0) = 1.
a(3) = 27 as a(2)^2 + 3^2 = 64 + 9 = 73 and 73 + 27 = 100 = 10^2.  Note that although 73 + 8 = 81 = 9^2, 8 cannot be chosen as a(2) = 8.
a(4) = 39 as a(3)^2 + 4^2 = 729 + 16 = 745, and 745 + 39 = 784 = 28^2 is the next smallest square.

Cf. A347594, A000290, A103605, A009000, A005408, A347754.

A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 5, 12, 13, 9, 12, 15, 8, 15, 17, 12, 16, 20, 7, 24, 25, 10, 24, 26, 15, 20, 25, 20, 21, 29, 18, 24, 30, 16, 30, 34, 12, 35, 37, 21, 28, 35, 9, 40, 41, 15, 36, 39, 24, 32, 40, 27, 36, 45, 14, 48, 50, 20, 48, 52, 24, 45, 51, 30, 40, 50, 28, 45, 53, 11, 60, 61, 33, 44, 55

Offset: 1

Hugo Pfoertner, Sep 29 2024

			   Triangle
   |  Perimeter
   |       x   y   z
   1  12 [ 3,  4,  5]
   2  24 [ 6,  8, 10]
   3  30 [ 5, 12, 13]
   4  36 [ 9, 12, 15]
   5  40 [ 8, 15, 17]
   6  48 [12, 16, 20]
   7  56 [ 7, 24, 25]
   8  60 [10, 24, 26]
   9  60 [15, 20, 25]
  10  70 [20, 21, 29]

Cf. A010814, A024364, A103605.

A374597 uses this order of sides.

A105521 Sums of area and perimeter of primitive Pythagorean triples.

Original entry on oeis.org

18, 60, 100, 140, 270, 280, 294, 462, 648, 728, 756, 1078, 1080, 1210, 1496, 1530, 1584, 1768, 2028, 2090, 2574, 2772, 2860, 2990, 3150, 3588, 3910, 4550, 4624, 4680, 4950, 5434, 5670, 5984, 6498, 6960, 7140, 7548, 8330, 8398, 8432, 8436, 8820, 9568, 10098

Offset: 1

Alexandre Wajnberg, May 02 2005

Douglas Butler, Table of Pythagorean triples up to 2100, Oundle School iCT Training Centre
Ron Knott, Pythagorean triples.

Cf. A103605, A103606, A024364, A024406, A105520

Corrected and extended by Harvey P. Dale, Oct 27 2018

A340178 Euler brick triples, side dimensions (a,b,c) in increasing order for a.

Original entry on oeis.org

44, 117, 240, 85, 132, 720, 88, 234, 480, 132, 351, 720, 140, 480, 693, 160, 231, 792, 170, 264, 1440, 176, 468, 960, 187, 1020, 1584, 195, 748, 6336, 220, 585, 1200, 240, 252, 275, 255, 396, 2160, 264, 702, 1440, 280, 960, 1386, 308, 819, 1680, 320, 462, 1584

Offset: 1

Ctibor O. Zizka, Dec 30 2020

How are primitive Euler bricks distributed in the (a,b,c) parameter space?

Robin Visser, Table of n, a(n) for n = 1..10000
Oliver Knill, Treasure Hunting Perfect Euler bricks, Math table, February 24, 2009.
Randall L. Rathbun, The Integer Cuboid Table, arXiv:1705.05929 [math.NT], 2017-2020.

Cf. A031173, A031174, A031175, A103605.

cp's OEIS Frontend

A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

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A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

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Examples

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A105521 Sums of area and perimeter of primitive Pythagorean triples.

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A340178 Euler brick triples, side dimensions (a,b,c) in increasing order for a.

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