A062775 -id:A062775 - cp's OEIS frontend

A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 4, 4, 6, 6, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 16, 18, 18, 18, 20, 22, 22, 22, 22, 24, 26, 26, 28, 28, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 36, 40, 42, 44, 46, 46, 48, 48, 48, 50, 50, 52, 54, 54, 54, 54, 62, 62, 62, 64, 64, 66, 66, 66, 68, 70

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001

nonn

			For n = 5 the Pythagorean triples are (3, 4, 5) and (4, 3, 5), so a (5) = 2.
For n = 10 the Pythagorean triples are (3, 4, 5), (4, 3, 5), (6, 8, 10) and (8, 6, 10), so a(10) = 4.
For n = 17 the Pythagorean triples are (3, 4, 5), (4, 5, 3), (5, 12, 13), (12, 5, 13), (6, 8, 10), (8, 6, 10), (8, 15, 17), (15, 8, 17), (9, 12, 15) and (12, 9, 15), so a(17) = 10.

Marius A. Burtea, Table of n, a(n) for n = 1..1000

Cf. A062775, A211422.

a(n) = 2*partial sums of A046080(n).

[#[: x in [1..n], y in [1..n]| IsSquare(x^2+y^2) and Floor(Sqrt(x^2+y^2)) le n]:n in [1..74]]; // Marius A. Burtea, Jan 22 2020

nq[n_] := SquaresR[2, n^2]/4 - 1; Accumulate@ Array[nq, 80] (* Giovanni Resta, Jan 23 2020 *)

Corrected and extended by Vladeta Jovovic, Jul 28 2001

A091143 Number of Pythagorean triples mod 2^n; i.e., number of solutions to x^2 + y^2 = z^2 mod 2^n.

Original entry on oeis.org

1, 4, 24, 96, 448, 1792, 7680, 30720, 126976, 507904, 2064384, 8257536, 33292288, 133169152, 534773760, 2139095040, 8573157376, 34292629504, 137304735744, 549218942976, 2197949513728, 8791798054912, 35175782154240, 140703128616960, 562881233944576

Offset: 0

T. D. Noe, Dec 22 2003

This Mathematica program is much more efficient than the one given in A062775.

T. D. Noe, Table of n, a(n) for n = 0..100
L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.
Index entries for linear recurrences with constant coefficients, signature (4,8,-32).

Cf. A062775 (number of Pythagorean triples mod n).

seq(op([(2^k-1)*2^(3*k-3),(2^k-1)*2^(3*k-1)]),k=1..30); # Robert Israel, Dec 03 2017

Table[n = 2^k; b = Table[0, {n}]; Do[ b[[1 + Mod[i^2, n]]]++, {i, 0, n - 1}]; cnt = 0; Do[m = x^2 + y^2; cnt = cnt + b[[1 + Mod[m, n]]], {x, 0, n - 1}, {y, 0, n - 1}]; cnt, {k, 0, 13}]

Vec(1/((4*x-1)*(8*x^2-1)) + O(x^100)) \\ Colin Barker, Oct 27 2013

a(2*k) = (2^(k+1)-1)*2^(3*k), a(2*k-1) = (2^k-1)*2^(3*k-1).
From Colin Barker, Oct 27 2013: (Start)
a(n) = 4*a(n-1) + 8*a(n-2) - 32*a(n-3).
G.f.: 1 / ((4*x-1)*(8*x^2-1)). (End)

A368197 Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

Original entry on oeis.org

1, 4, 4, 18, 0, 9, 32, 8, 0, 24, 100, 0, 0, 0, 25, 72, 72, 36, 0, 0, 36, 294, 0, 0, 0, 0, 0, 49, 256, 64, 0, 96, 0, 0, 0, 96, 486, 0, 144, 0, 0, 0, 0, 0, 99, 400, 400, 0, 0, 100, 0, 0, 0, 0, 100, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121

Offset: 1

Mats Granvik, Dec 16 2023

Row n has sum n^3. The number of nonzero terms in row n appears to be A000005(n). It appears that Sum_{k=1..n} T(n,k)*A023900(k) = A063524(n). Main diagonal appears to be A062775. First column appears to be A053191.

It appears that when p > 2 in f(x,y,z,p) = x^p + y^p - z^p and T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z,p), n) = k], then Sum_{k=1..n} T(n,k)*A023900(k) is not equal to A063524(n). - Mats Granvik, May 07 2024

			Triangle begins:
     1;
     4,   4;
    18,   0,   9;
    32,   8,   0,  24;
   100,   0,   0,   0,  25;
    72,  72,  36,   0,   0,  36;
   294,   0,   0,   0,   0,   0,  49;
   256,  64,   0,  96,   0,   0,   0,  96;
   486,   0, 144,   0,   0,   0,   0,   0,  99;
   400, 400,   0,   0, 100,   0,   0,   0,   0, 100;
  1210,   0,   0,   0,   0,   0,   0,   0,   0,   0, 121;
  ...

Cf. A000578, A000005, A023900, A063524, A062775, A053191, A367689.

nn = 11; p = 2; f = x^p + y^p - z^p; Flatten[Table[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 1, nn}]]

T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

A096020 Number of Pythagorean quintuples mod n; i.e., number of solutions to v^2 + w^2 + x^2 + y^2 = z^2 mod n.

Original entry on oeis.org

1, 16, 81, 192, 625, 1296, 2401, 3072, 6723, 10000, 14641, 15552, 28561, 38416, 50625, 47104, 83521, 107568, 130321, 120000, 194481, 234256, 279841, 248832, 393125, 456976, 544563, 460992, 707281, 810000, 923521, 753664

Offset: 1

T. D. Noe, Jun 15 2004

			x + 16 x^2 + 81 x^3 + 192 x^4 + 625 x^5 + 1296 x^6 + 2401 x^7 + ...

L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.

Cf. A062775 (number of solutions to x^2 + y^2 = z^2 mod n), A096018 (number of solutions to w^2 + x^2 + y^2 = z^2 mod n).

Table[cnt=0; Do[If[Mod[v^2+w^2+x^2+y^2-z^2, n]==0, cnt++ ], {v, 0, n-1}, {w, 0, n-1}, {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 30}]
a[ n_] := If[ n < 1, 0, Sum[ 1 - Sign[ Mod[ v^2 + w^2 + x^2 + y^2 - z^2, n]], {v, n}, {w, n}, {x, n}, {y, n}, {z, n}]]; (* Michael Somos, Jan 21 2012 *)

A331996 Number of Pythagorean triples mod n: total number of solutions (x,y,z) to x^2 + y^2 = z^2 mod n with x <= y.

Original entry on oeis.org

1, 3, 5, 14, 13, 19, 31, 52, 54, 51, 61, 110, 85, 111, 113, 232, 161, 207, 181, 302, 227, 243, 287, 436, 375, 339, 450, 614, 421, 451, 511, 912, 545, 611, 619, 1206, 685, 723, 761, 1204, 881, 895, 925, 1454, 1242, 1103, 1151, 2024, 1414, 1475, 1317, 2030, 1405

Offset: 1

Yinxi Pan, Feb 03 2020

nonn

Based on A062775, but that sequence counts (x,y,z) and (y,x,z) as different pairs.

			Below is an example for n = 3 (a(3) = 5).
(0 0 0)
(1 0 1)
(1 0 2)
(2 0 1)
(2 0 2)
In contrast, A062775, counts (1 0 1) and (0 1 1), etc. as different pairs and therefore A062775(3) = 9 .

Cf. A062775.

a[n_] := Block[{q = Association[(#[[1]] -> #[[2]]) & /@ Tally[ Mod[ Range[ n]^2, n]]]}, Sum[ Lookup[q, Mod[x^2 + y^2, n], 0], {x,n}, {y,x}]]; Array[a, 53] (* Giovanni Resta, Feb 04 2020 *)

More terms from Giovanni Resta, Feb 04 2020

cp's OEIS Frontend

A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

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A091143 Number of Pythagorean triples mod 2^n; i.e., number of solutions to x^2 + y^2 = z^2 mod 2^n.

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A368197 Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

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A096020 Number of Pythagorean quintuples mod n; i.e., number of solutions to v^2 + w^2 + x^2 + y^2 = z^2 mod n.

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Mathematica

A331996 Number of Pythagorean triples mod n: total number of solutions (x,y,z) to x^2 + y^2 = z^2 mod n with x <= y.

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