A198409 -id:A198409 - cp's OEIS frontend

A198385 Second of a triple of squares in arithmetic progression.

25, 100, 169, 225, 289, 400, 625, 676, 625, 841, 900, 1156, 1369, 1225, 1681, 1521, 1600, 2500, 2025, 2704, 2601, 2500, 3721, 2809, 3025, 4225, 3364, 3600, 4225, 4225, 4225, 4624, 5625, 5476, 7225, 4900, 6724, 6084, 5329, 5625, 6400, 7225, 7225, 7225, 7921

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198385 n = a198385_list !! (n-1)
a198385_list = map (^ 2) a198389_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 2]] (* Jean-François Alcover, Oct 19 2021 *)

a(n) = A198389(n)^2.
a(n) - A198384(n) = A198386(n) - a(n) = A198387(n).
A198436(n) = a(A198409(n)).

A198384 First of a triple of squares in arithmetic progression.

Original entry on oeis.org

1, 4, 49, 9, 49, 16, 289, 196, 25, 1, 36, 196, 529, 49, 961, 441, 64, 1156, 81, 784, 441, 100, 2401, 289, 121, 2209, 4, 144, 1225, 529, 169, 784, 2601, 2116, 5041, 196, 3844, 1764, 49, 225, 256, 1681, 1225, 289, 1681, 2401, 6241, 9, 4624, 324, 9409, 361

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198384 n = a198384_list !! (n-1)
a198384_list = map (^ 2) a198388_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 19 2021 *)

a(n) = A198388(n)^2.
A198385(n) - a(n) = A198386(n) - A198385(n) = A198387(n).
A198435(n) = a(A198409(n)).

Thanks to Benoit Jubin, who had the idea for sequences A198384 .. A198390 and A198435 .. A198441.

A198386 Third of a triple of squares in arithmetic progression.

Original entry on oeis.org

49, 196, 289, 441, 529, 784, 961, 1156, 1225, 1681, 1764, 2116, 2209, 2401, 2401, 2601, 3136, 3844, 3969, 4624, 4761, 4900, 5041, 5329, 5929, 6241, 6724, 7056, 7225, 7921, 8281, 8464, 8649, 8836, 9409, 9604, 9604, 10404, 10609, 11025, 12544, 12769, 13225

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198386 n = a198386_list !! (n-1)
a198386_list = map (^ 2) a198390_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]] ][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 3]] (* Jean-François Alcover, Oct 19 2021 *)

a(n) = A198390(n)^2.
a(n) - A198385(n) = A198385(n) - A198384(n) = A198387(n).
A198437(n) = a(A198409(n)).

A198441 Square root of third term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

Original entry on oeis.org

7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 119, 127, 137, 151, 161, 161, 167, 191, 193, 199, 217, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 287, 289, 311, 313, 329, 329, 337, 343, 353, 359, 367, 383, 391, 391, 401, 409, 431, 433

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

This sequence gives the sum of the two legs (catheti) x + y of primitive Pythagorean triangles (x,y,z) with y even and gcd(x,y) = 1, ordered nondecreasingly (with multiple entries). See A058529(n), n>=2, for the sequence without multiple entries. For the proof, put in the Zumkeller link w = x + y, v = z and u = abs(x - y). This works because w^2 - v^2 = v^2 - u^2, hence u^2 = 2*v^2 - w^2 = 2*z^2 - (x+y)^2 = 2*(x^2 + y^2) - (x+y)^2 = x^2 + y^2 - 2*x*y = (x-y)^2. The primitivity of the arithmetic progression triples follows from the one of the Pythagorean triples: gcd(u,w) = 1 follows from gcd(x,y) = 1, then gcd(u,v,w) = gcd(gcd(u,w),v) = 1. The converse can also be proved: given a primitive arithmetic progression triple (u,v,w), 1 <= u < v < w, gcd(u,v,w) = 1, the corresponding primitive Pythagorean triple with even y is ((w-u)/2,(w+u)/2,v) or ((w+u)/2,(w-u)/2,v), depending on whether (w+u)/2 is even or odd, respectively. - Wolfdieter Lang, May 22 2013

n appears A330174(n) times. - Ray Chandler, Feb 26 2020

			Primitive Pythagorean triangle connection: a(1) = 7 because (u,v,w) = (1,5,7) corresponds to the primitive Pythagorean triangle (x = (w-u)/2, y = (w+u)/2, z = v) = (3,4,5) with leg sum 3 + 4 = 7. - _Wolfdieter Lang_, May 23 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A225949 (triangle version of leg sums).
Cf. A198384, A198385, A198386.
Cf. A058529, A198390, A198409, A198437, A330174.

a198441 n = a198441_list !! (n-1)
a198441_list = map a198390 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]][[All, 3]] (* Jean-François Alcover, Oct 22 2021 *)

A198437(n) = a(n)^2; a(n) = A198390(A198409(n)).

A198387 Common differences in triples of squares in arithmetic progression.

Original entry on oeis.org

24, 96, 120, 216, 240, 384, 336, 480, 600, 840, 864, 960, 840, 1176, 720, 1080, 1536, 1344, 1944, 1920, 2160, 2400, 1320, 2520, 2904, 2016, 3360, 3456, 3000, 3696, 4056, 3840, 3024, 3360, 2184, 4704, 2880, 4320, 5280, 5400, 6144, 5544, 6000, 6936, 6240, 5880

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198387 n = a198387_list !! (n-1)
a198387_list = zipWith (-) a198385_list a198384_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
#[[2]] - #[[1]]& /@ Flatten[DeleteCases[triples /@ Range[wmax], {}] , 2] (* Jean-François Alcover, Oct 21 2021 *)

a(n) = A198385(n) - A198384(n) = A198386(n) - A198385(n).

A198438(n) = a(A198409(n)).

A198435 First term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

Original entry on oeis.org

1, 49, 49, 289, 1, 529, 961, 2401, 289, 2209, 529, 5041, 49, 1681, 1681, 6241, 9409, 49, 961, 5329, 16129, 14161, 7921, 289, 25921, 2209, 12769, 27889, 14161, 1, 39601, 2401, 5329, 10609, 25921, 49729, 58081, 529, 961, 10609, 7921, 36481, 82369, 22801, 47089

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198384, A198409, A198437, A198438, A198439

a198435 n = a198435_list !! (n-1)
a198435_list = map a198384 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

a(n) = A198439(n)^2 = A198384(A198409(n));

A198436(n) - a(n) = A198437(n) - A198436(n) = A198438(n).

A198390 Square root of third term of a triple of squares in arithmetic progression.

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 98, 102, 103, 105, 112, 113, 115, 119, 119, 119, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

There is a connection to the leg sums of Pythagorean triangles.
See a comment on the primitive case under A198439, which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
n appears A331671(n) times. - Ray Chandler, Feb 26 2020

			Connection to leg sums of Pythagorean triangles: a(2) = 14 because (in the notation of the Zumkeller link) (u,v,w)= (2,10,14) = 2*(1,5,7), and this corresponds to the non-primitive Pythagorean triangle 2*(x=(7-1)/1,y=(1+7)/2,z=5) = 2*(3,4,5) with leg sum 2*(3+4) = 14. - _Wolfdieter Lang_, May 23 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198386, A198409, A198439, A198441, A331671.

a198390 n = a198390_list !! (n-1)
a198390_list = map (\(,,x) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 3]] (* Jean-François Alcover, Oct 20 2021 *)

is(n)=my(t=n^2);forstep(i=2-n%2,n-2,2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

A198386(n) = a(n)^2.

A198441(n) = a(A198409(n)).

A198436 Second term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384, A198385, A198386).

Original entry on oeis.org

25, 169, 289, 625, 841, 1369, 1681, 3721, 2809, 4225, 4225, 7225, 5329, 7225, 7921, 10201, 12769, 9409, 11881, 15625, 21025, 21025, 22201, 18769, 32761, 24649, 29929, 38809, 34225, 28561, 48841, 34225, 37249, 42025, 52441, 66049, 70225, 42025, 48841, 54289

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198385, A198409, A198435, A198437, A198438, A198440.

a198436 n = a198436_list !! (n-1)
a198436_list = map a198385 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 2]] (* Jean-François Alcover, Oct 20 2021 *)

a(n) = A198440(n)^2 = A198385(A198409(n)).

a(n) - A198435(n) = A198437(n) - a(n) = A198438(n).

A198437 Third term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

Original entry on oeis.org

49, 289, 529, 961, 1681, 2209, 2401, 5041, 5329, 6241, 7921, 9409, 10609, 12769, 14161, 14161, 16129, 18769, 22801, 25921, 25921, 27889, 36481, 37249, 39601, 47089, 47089, 49729, 54289, 57121, 58081, 66049, 69169, 73441, 78961, 82369, 82369, 83521, 96721

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198435, A198436, A198438, A198409, A198441.

a198437 n = a198437_list !! (n-1)
a198437_list = map a198386 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 3]] (* Jean-François Alcover, Oct 20 2021 *)

a(n) = A198441(n)^2 = A198386(A198409(n));

a(n) - A198436(n) = A198436(n) - A198435(n) = A198438(n).

A198438 Common differences in triples of squares in arithmetic progression, that are not a multiples of other triples in (A198384, A198385, A198386).

Original entry on oeis.org

24, 120, 240, 336, 840, 840, 720, 1320, 2520, 2016, 3696, 2184, 5280, 5544, 6240, 3960, 3360, 9360, 10920, 10296, 4896, 6864, 14280, 18480, 6840, 22440, 17160, 10920, 20064, 28560, 9240, 31824, 31920, 31416, 26520, 16320, 12144, 41496, 47880, 43680, 50160

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198438 n = a198438_list !! (n-1)
a198438_list = map a198387 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
#[[2]] - #[[1]]& /@ DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]] (* Jean-François Alcover, Oct 22 2021 *)

a(n) = A198387(A198409(n)) = A198436(n) - A198435(n) = A198437(n) - A198436(n).

cp's OEIS Frontend

A198385 Second of a triple of squares in arithmetic progression.

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A198384 First of a triple of squares in arithmetic progression.

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A198386 Third of a triple of squares in arithmetic progression.

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A198441 Square root of third term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

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A198387 Common differences in triples of squares in arithmetic progression.

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A198435 First term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

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A198390 Square root of third term of a triple of squares in arithmetic progression.

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A198436 Second term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384, A198385, A198386).

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