A198388 -id:A198388 - cp's OEIS frontend

A198384 First of a triple of squares in arithmetic progression.

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.

A198409 Positions in sequences A198384, A198385 and A198386 to indicate triples of squares in arithmetic progression, that are not multiples of earlier triples.

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 15, 23, 24, 26, 30, 35, 39, 42, 45, 47, 51, 54, 62, 69, 70, 72, 83, 84, 88, 97, 98, 102, 107, 114, 115, 124, 126, 129, 136, 141, 142, 143, 156, 157, 167, 169, 172, 177, 181, 188, 191, 201, 205, 208, 214, 218, 229, 230, 237, 244, 249, 253

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

A198435(n) = A198384(a(n)); A198439(n) = A198388(a(n));
A198436(n) = A198385(a(n)); A198440(n) = A198389(a(n));
A198437(n) = A198386(a(n)); A198441(n) = A198390(a(n));
A198438(n) = A198387(a(n)).

Ray Chandler, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Table of initial values

import Data.List (elemIndices)
a198409 n = a198409_list !! (n-1)
a198409_list = map (+ 1) $ elemIndices 1 $ map a008966 $
   zipWith gcd a198384_list $ zipWith gcd a198385_list a198386_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];
Position[tt, t_List /; SquareFreeQ[GCD@@t]] // Flatten (* Jean-François Alcover, Oct 24 2021 *)

A008966(A050873(A198439(n),A050873(A198440(n),A050873(A198441(n))))) = 1.

A198439 Square root of first 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

1, 7, 7, 17, 1, 23, 31, 49, 17, 47, 23, 71, 7, 41, 41, 79, 97, 7, 31, 73, 127, 119, 89, 17, 161, 47, 113, 167, 119, 1, 199, 49, 73, 103, 161, 223, 241, 23, 31, 103, 89, 191, 287, 151, 217, 287, 137, 233, 71, 337, 79, 137, 17, 281, 359, 391, 49, 113, 119, 217

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

This sequence gives the values |x-y| of primitive Pythagorean triangles (x,y,z) with even y ordered according to the nondecreasing values of the leg sums x+y (called w in the Zumkeller link, and given in A198441). For the equivalence to primitive Pythagorean triples with even y see a comment in A198441. - Wolfdieter Lang, May 22 2013

			From _Wolfdieter Lang_, May 22 2013: (Start)
Primitive Pythagorean triple (x,y,z), y even, connection:
a(2) = 7 because the triple with second smallest leg sum x+y = 17 = A198441(2) is (5,12,13), and |x - y| = y - x = 12 - 5 = 7.
a(3) = 7 because x + y = A198441(3) = 23, (x,y,z) = (15,8,17) (the primitive triple with third smallest leg sum), and |x-y| = x - y = 15 - 8 = 7. (End)

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

a198439 n = a198439_list !! (n-1)
a198439_list = map a198388 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, 1]] (* Jean-François Alcover, Oct 22 2021 *)

A198435(n) = a(n)^2; a(n) = A198388(A198409(n)).

A198389 Square root of second term of a triple of squares in arithmetic progression.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 25, 29, 30, 34, 37, 35, 41, 39, 40, 50, 45, 52, 51, 50, 61, 53, 55, 65, 58, 60, 65, 65, 65, 68, 75, 74, 85, 70, 82, 78, 73, 75, 80, 85, 85, 85, 89, 91, 101, 87, 100, 90, 113, 95, 104, 97, 102, 100, 111, 122, 106, 105, 123, 109

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Apart from its initial 1, A001653 is a subsequence: for all n>1 exists an m such that A198388(m)=1 and a(m)=A001653(n). [observed by Zak Seidov, Reinhard Zumkeller, Oct 25 2011]

There is a connection to hypotenuses of Pythagorean triangles. See a comment for the primitive case on A198441 which applies here mutatis mutandis. - Wolfdieter Lang, May 23 2013

			Connection to Pythagorean triangle hypotenuses: a(20) = 10 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the Pythagorean triangle is 2*(x=(7-1)/2,y=(1+7)/2,5) = 2*(3,4,5) with hypotenuse 2*5 = 10. - _Wolfdieter Lang_, May 23 2013

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

Cf. A198385, A198409, A198440.

a198389 n = a198389_list !! (n-1)
a198389_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, 2]] (* Jean-François Alcover, Oct 20 2021 *)

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

A198440(n) = a(A198409(n)).

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A198384 First of a triple of squares in arithmetic progression.

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A198409 Positions in sequences A198384, A198385 and A198386 to indicate triples of squares in arithmetic progression, that are not multiples of earlier triples.

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A198439 Square root of first 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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A198389 Square root of second term of a triple of squares in arithmetic progression.

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