A198385 -id:A198385 - cp's OEIS frontend

A198386 Third of a triple of squares in arithmetic progression.

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)).

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)).

A292313 Numbers that are the sum of three squares in arithmetic progression.

Original entry on oeis.org

75, 300, 507, 675, 867, 1200, 1875, 2028, 2523, 2700, 3468, 3675, 4107, 4563, 4800, 5043, 6075, 7500, 7803, 8112, 8427, 9075, 10092, 10800, 11163, 12675, 13872, 14700, 15987, 16428, 16875, 18252, 19200, 20172, 21675, 22707, 23763, 24300, 24843, 27075, 28227, 30000, 30603

Offset: 1

Antonio Roldán, Sep 14 2017

nonn

			75 = 1^2 + 5^2 + 7^2 = 1 + 25 + 49, with 25 - 1 = 49 - 25 = 24.
675 = 3^2 + 15^2 + 21^2 = 9 + 225 + 441, with 225 - 9 = 441 - 225 = 216.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subsequence of A033428.

Cf. A000290, A000378, A004431, A134422, A292309, A292310, A292314, A292316.

t=4; k=3; while(t<=13000, i=k; e=0; v=t+i; while(i>1&&e==0, if(issquare(v), m=3*t; e=1; print1(m,", ")); i+=-2; v+=i); k+=2; t+=k)

Sequence is 3*(distinct elements in A198385).

Numbers of the form 3*m^2 where 2*m^2 is in A004431. - Chai Wah Wu, Oct 05 2017

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)).

A230479 Integer areas of the integer-sided triangles such that the length of the circumradius is a square.

Original entry on oeis.org

168, 336, 432, 600, 768, 2688, 5376, 6000, 6912, 9600, 12288, 13608, 14280, 20280, 27216, 28560, 30720, 32928, 34560, 34992, 38640, 43008, 46200, 48600, 62208, 69360, 77280, 86016, 96000, 105000, 108000, 110592, 118272, 153600, 196608

Offset: 1

Michel Lagneau, Oct 20 2013

nonn

The primitive areas are 168, 338, 432, 600, 768, 13608, 14280, 20280, 27216, ...
The non-primitive areas 16*a(n) are in the sequence because if R is the circumradius corresponding to a(n), then 4*R is the circumradius corresponding to 16*a(n).
Each circumradius belongs to the sequence {25, 100, 169, 225, 289, 400, 625, 676, ...}, and it seems that this last sequence is A198385 (second of a triple of squares in arithmetic progression).
The following table gives the first values (A, R, a, b, c) where A is the integer area, R the radius of the circumcircle, and a, b, c are the integer sides of the triangle.
**************************************
*     A  *    R  *   a  *   b  *  c  *
**************************************
*   168  *   25  *  14  *  30 *  40  *
*   336  *   25  *  14  *  48 *  50  *
*   432  *   25  *  30  *  30 *  48  *
*   600  *   25  *  30  *  40 *  50  *
*   768  *   25  *  40  *  40 *  48  *
*  2688  *  100  *  56  * 120 * 160  *
*  5376  *  100  *  56  * 192 * 200  *
*  6912  *  100  * 120  * 120 * 192  *
*  9600  *  100  * 120  * 160 * 200  *
* 12288  *  100  * 160  * 160 * 192  *
* 13608  *  225  * 126  * 270 * 360  *
* 14280  *  169  * 130  * 238 * 312  *
* 20280  *  169  * 130  * 312 * 338  *
* 27216  *  225  * 126  * 432 * 450  *
.............................

			168 is in the sequence because the area of the triangle (14, 30, 40) is given by Heron's formula A = sqrt(42*(42-14)*(42-30)*(42-40))= 168 where the number 42 is the semiperimeter, and the circumcircle is given by R = a*b*c/(4*A) = 14*30*40/(4*168) = 25, which is a square.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric W. Weisstein, MathWorld: Circumradius

Cf. A188158, A208984, A210207.

nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[a*b*c/(4*Sqrt[area2])]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);

Circumradius R = a*b*c/4A.

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A198386 Third of a triple of squares in arithmetic progression.

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A198387 Common differences in triples of squares in arithmetic progression.

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A292313 Numbers that are the sum of three squares in arithmetic progression.

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

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A230479 Integer areas of the integer-sided triangles such that the length of the circumradius is a square.

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