A237516 -id:A237516 - cp's OEIS frontend

A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

12, 84, 312, 840, 1860, 3612, 6384, 10512, 16380, 24420, 35112, 48984, 66612, 88620, 115680, 148512, 187884, 234612, 289560, 353640, 427812, 513084, 610512, 721200, 846300, 987012, 1144584, 1320312, 1515540, 1731660, 1970112, 2232384, 2520012, 2834580

Offset: 1

Miguel-Ángel Pérez García-Ortega, May 31 2025

			Triangles begin:
  n=1:      5,   12,   13;
  n=2:     13,   84,   85;
  n=3:     25,  312,  313;
  ...
This sequence gives the middle column.

José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A002378 (inradius), A001844 (short leg), A008514 (sum of the legs), A237516 (semiperimeter), A384566 (area).

a(n) = 2 * A002378(n) * (A002378(n) + 1).

A384566 Area of the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 30, 546, 3900, 17220, 56730, 153510, 360696, 762120, 1482390, 2698410, 4652340, 7665996, 12156690, 18654510, 27821040, 40469520, 57586446, 80354610, 110177580, 148705620, 197863050, 259877046, 337307880, 433080600, 550518150, 693375930, 865877796, 1072753500, 1319277570, 1611309630

Offset: 0

Miguel-Ángel Pérez García-Ortega, Jun 03 2025

a(n) is multiple of 6 for all n.

			For n=1, the short leg is A384288(1,1) = 5 and the long leg is A384288(1,2) = 12 so the area is then a(1) = (5 * 12 )/2 = 30.

José Miguel Blanco Casado and Miguel Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A002378, A384288, A008514, A237516.

a=Table[(n(n+1)),{n,0,30}];Apply[Join,Map[{#(#+1)(2#+1)}&,a]]

a(n) = (A384288(n,1) * A384288(n,2))/2.

a(n) = A002378(n)*(A002378(n) + 1)*(2*A002378(n) + 1).

cp's OEIS Frontend

A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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