A227481 -id:A227481 - cp's OEIS frontend

A009023 Long legs of Pythagorean triangles.

4, 8, 12, 15, 16, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 64, 68, 70, 72, 75, 76, 77, 80, 84, 88, 90, 91, 92, 96, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 126, 128, 132, 135, 136, 140, 143, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

David W. Wilson

nonn

A227481(a(n)) > 1. - Reinhard Zumkeller, Oct 11 2013
This is A009012 (sorted A046084) without duplicates. - Andrey Zabolotskiy, Dec 27 2017
Does a(n)/n converge to some limit? - Benoit Cloitre, Oct 18 2009
For n = {52000, 72000, 100000}, n/a(n) = {0.499, 0.50175, 0.50428}. - Alex Ratushnyak, Jan 17 2019

Wacław Sierpiński, Pythagorean triangles, Dover books. [Benoit Cloitre, Oct 17 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A074235 (complement), A009012, A046084, A227481.

a009023 n = a009023_list !! (n-1)
a009023_list = filter ((> 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

A069011 Triangle with T(n,k) = n^2 + k^2.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200

Offset: 0

Henry Bottomley, Apr 02 2002

For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
A227481(n) = number of squares in row n. - Reinhard Zumkeller, Oct 11 2013
Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - Stefano Spezia, Aug 07 2025

			Triangle T(n,k) begins:
    0;
    1,   2;
    4,   5,   8;
    9,  10,  13,  18;
   16,  17,  20,  25,  32;
   25,  26,  29,  34,  41,  50;
   36,  37,  40,  45,  52,  61,  72;
   49,  50,  53,  58,  65,  74,  85,  98;
   64,  65,  68,  73,  80,  89, 100, 113, 128;
   81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
  100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
Column k=0 gives A000290.
Main diagonal gives A001105.
Row sums give A132124.
T(2n,n) gives A033429.

a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
   f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
-- Reinhard Zumkeller, Oct 11 2013

Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 07 2025 *)

T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013

G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Aug 04 2025

A074235 Numbers that cannot be a long leg of an integer right triangle.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 81, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 98, 101, 102, 103, 106, 107, 109, 111

Offset: 1

Zak Seidov, Sep 18 2002

nonn

The possible values of a long leg of an integer right triangle are in A009012.

A227481(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

			5 is a term because a^2 + 5^2 = c^2 has no solution for a < 5 with integers a, c.
13 is a term because a^2 + 13^2 = c^2 has no solution for a < 13 with integers a, c.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A009012.

Cf. A009023 (complement).

a074235 n = a074235_list !! (n-1)
a074235_list = filter ((== 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

cp's OEIS Frontend

A009023 Long legs of Pythagorean triangles.

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A069011 Triangle with T(n,k) = n^2 + k^2.

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A074235 Numbers that cannot be a long leg of an integer right triangle.

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