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URL: https://oeis.org/A384566

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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.
3
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
0,2
COMMENTS
a(n) is multiple of 6 for all n.
LINKS
José Miguel Blanco Casado and Miguel Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.
FORMULA
a(n) = (A384288(n,1) * A384288(n,2))/2.
a(n) = A002378(n)*(A002378(n) + 1)*(2*A002378(n) + 1).
From Andrew Howroyd, Nov 12 2025: (Start)
a(n) = n*(n + 1)*(n^2 + n + 1)*(2*n^2 + 2*n + 1).
G.f.: 6*x*(5 + 56*x + 118*x^2 + 56*x^3 + 5*x^4)/(1 - x)^7. (End)
EXAMPLE
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.
MATHEMATICA
a=Table[(n(n+1)), {n, 0, 30}]; Apply[Join, Map[{#(#+1)(2#+1)}&, a]]
CROSSREFS
KEYWORD
nonn,easy
STATUS
approved