A338275 -id:A338275 - cp's OEIS frontend

A338895 Three-column table read by rows giving Pythagorean triples [a,b,c] that are the (constant) differences between consecutive triples in rows of A338275.

0, 4, 4, 6, 8, 10, 0, 16, 16, 16, 12, 20, 10, 24, 26, 0, 36, 36, 30, 16, 34, 24, 32, 40, 14, 48, 50, 0, 64, 64, 48, 20, 52, 42, 40, 58, 32, 60, 68, 18, 80, 82, 0, 100, 100, 70, 24, 74, 64, 48, 80, 54, 72, 90, 40, 96, 104, 22, 120, 122, 0, 144, 144

Offset: 1

David Lovler, Nov 14 2020

			The table begins:
  [ 0,   4,   4],
  [ 6,   8,  10],
  [ 0,  16,  16],
  [16,  12,  20],
  [10,  24,  26],
  [ 0,  36,  36],
  [30,  16,  34],
  [24,  32,  40],
  [14,  48,  50],
  [ 0,  64,  64],
  [48,  20,  52],
  [42,  40,  58],
  [32,  60,  68],
  [18,  80,  82],
  [ 0, 100, 100],
  [70,  24,  74],
  [64,  48,  80],
  [54,  72,  90],
  [40,  96, 104],
  [22, 120, 122],
  [ 0, 144, 144],
  ...

David Lovler, Table of n, a(n) for n = 1..300
David Lovler, Triples for m up to 100

Cf. A338275, A338896.

Table[{((#1 - 1)^2 - #2^2)/2, (#1 - 1) #2, ((#1 - 1)^2 + #2^2)/2} & @@ {m, n}, {m, 3, 13, 2}, {n, 2, m, 2}] // Flatten (* Michael De Vlieger, Dec 04 2020 *)

lista(mm) = {forstep (m=3, mm, 2, forstep (n=2, m, 2, print([((m-1)^2 - n^2)/2, (m-1)*n, ((m-1)^2 + n^2)/2]);););} \\ Michel Marcus, Dec 04 2020

a = ((m-1)^2 - n^2)/2, b = (m-1)*n, c = ((m-1)^2 + n^2)/2, where m and n generate the A338275 row in question.

A338896 Inradii of Pythagorean triples of A338895.

Original entry on oeis.org

0, 2, 0, 4, 4, 0, 6, 8, 6, 0, 8, 12, 12, 8, 0, 10, 16, 18, 16, 10, 0, 12, 20, 24, 24, 20, 12, 0, 14, 24, 30, 32, 30, 24, 14, 0, 16, 28, 36, 40, 40, 36, 28, 16, 0, 18, 32, 42, 48, 50, 48, 42, 32, 18, 0, 20, 36, 48, 56, 60, 60, 56, 48, 36, 20, 0

Offset: 1

David Lovler, Nov 14 2020

Without the 0's, the sequence becomes 2*A003991. The 0 indices are triangular numbers A000217.

			m 3
n 2  [0,4,4]    0
------------------
m 5
n 2  [6,8,10]   2
n 4  [0,16,16]  0
------------------
m 7
n 2  [16,12,20] 4
n 4  [10,24,26] 4
n 6  [0,36,36]  0
------------------
m 9
n 2  [30,16,34] 6
n 4  [24,32,40] 8
n 6  [14,48,50] 6
n 8  [0,64,64]  0
------------------.
The 7th row of A338895 is [30,16,34], so a(7) = 30*16/(30+16+34) = 6.
As a triangle:
   0
   2, 0
   4, 4, 0
   6, 8, 6, 0
   8, 12, 12, 8, 0
  10, 16, 18, 16, 10, 0
  12, 20, 24, 24, 20, 12, 0,

Ron Knott, Pythagorean Triples and Online Calculators

Cf. A338275, A338895.

Cf. A003991 (see 2nd comment).

Table[#1 #2/Total[{##}] & @@ {((#1 - 1)^2 - #2^2)/2, (#1 - 1) #2, ((#1 - 1)^2 + #2^2)/2} & @@ {m, n}, {m, 3, 23, 2}, {n, 2, m, 2}] // Flatten (* Michael De Vlieger, Dec 04 2020 *)

lista(mm) = forstep (m=3, mm, 2, forstep (n=2, m, 2, my(v=[((m-1)^2 - n^2)/2, (m-1)*n, ((m-1)^2 + n^2)/2]); print1(v[1]*v[2]/vecsum(v), ", "))); \\ Michel Marcus, Dec 04 2020

When m and n define a row of triples in A338275 that gives rise to a triple (a row) of A338895, the current term corresponding to such a row is (m-n-1)*n/2.
If [a,b,c] is the n-th row of A338895, then a(n) = a*b/(a+b+c).
T(n, k) = 2*k*(n - k). Follows from the first comment. - Peter Luschny, Apr 17 2023

cp's OEIS Frontend

A338895 Three-column table read by rows giving Pythagorean triples [a,b,c] that are the (constant) differences between consecutive triples in rows of A338275.

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