A331671 -id:A331671 - cp's OEIS frontend

A118905 Sum of legs of Pythagorean triangles (without multiple entries).

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 102, 103, 105, 112, 113, 115, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 147, 151, 153, 154, 155, 158, 161, 164, 167, 168, 170, 175, 178, 182, 184, 186, 187, 188

Offset: 1

Giovanni Resta, May 05 2006

nonn

The prime numbers in this sequence define A001132 (see comment in A001132). - Richard Choulet, Dec 16 2008
For the sum of legs of Pythagorean triangles with multiple entries see A198390. - Wolfdieter Lang, May 24 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
For the sum of legs of primitive Pythagorean triangles see A120681. - Wolfdieter Lang, Feb 17 2015
n is in the sequence iff A331671(n) > 0. - Ray Chandler, Feb 26 2020

			7 = 3 + 4 and 3^2 + 4^2 = 5^2.
a(14) = 49 = 7^2 from the primitive Pythagorean triangle (x,y,z) = (9,40,41), and from the non-primitive one 7*(3,4,5); a(42) = 119 = 7*17 from four Pythagorean triangles (39,80,89) and (99,20,181) (both primitive) and 7*(5,12,13), 17*(3,4,5). - _Wolfdieter Lang_, May 24 2013

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A009096, A118903, A118904, A058529, A001132, A120681, A331671.

[m:m in [2..200]|#[k:k in [1..m-1]|IsSquare(k^2+(m-k)^2)] ne 0]; // Marius A. Burtea, Jul 29 2019

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

More terms from 147 on. - Richard Choulet, Nov 24 2009

Name specified. - Wolfdieter Lang, May 24 2013

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)).

A254064 Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 102, 103, 105, 112, 113, 115, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 151, 153, 154, 155, 158, 164, 167

Offset: 1

Colin Barker, Jan 24 2015

nonn

Equivalently positive integers whose square is expressible in exactly one way as -x^2 + 2xy + y^2 with 0 < x < y by replacing (x,y) with (2x,x+y). As such this sequence represents the sum of legs that are unique to a single Pythagorean triangle. - Ray Chandler, Feb 18 2020

n is in the sequence iff A331671(n)=1. - Ray Chandler, Feb 26 2020

			7 is in the sequence because the only solution to x^2 + 6xy + y^2 = 49 with 0 < x < y is (x,y) = (2,3).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 750 terms from Colin Barker)

Cf. A084645, A232437, A248599, A254063, A331671.

s[n_] := Solve[0 < x < y && n^2 == x^2 + 6 x y + y^2, {x, y}, Integers];
Reap[For[n = 1, n < 200, n++, If[Length[s[n]]==1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)

A374061 Number of solutions to a^2 + n = 2 * b^2 with 0 < a < b.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Seiichi Manyama, Jun 27 2024

			3^2 + 119 = 2 * 8^2 and 9^2 + 119 = 2 * 10^2. So a(119) = 2.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A331671.

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A118905 Sum of legs of Pythagorean triangles (without multiple entries).

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

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A254064 Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.

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Mathematica

A374061 Number of solutions to a^2 + n = 2 * b^2 with 0 < a < b.

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