A385187 Area of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A002378(n) and its long leg and hypotenuse are consecutive natural numbers.

0, 6, 330, 3036, 14820, 51330, 142926, 341880, 731016, 1433790, 2625810, 4547796, 7519980, 11957946, 18389910, 27475440, 40025616, 57024630, 79652826, 109311180, 147647220, 196582386, 258340830, 335479656, 430920600, 547983150, 690419106, 862448580, 1068797436, 1314736170, 1606120230

Offset

0,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (A385022(n,1) * A385022(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*(1 + 48*x + 142*x^2 + 48*x^3 + x^4)/(1 - x)^7. (End)

Example

For n=2, the short leg is A385022(2,1) = 11 and the long leg is A385022(2,2) = 60  so the area is then a(2) = (11 * 60)/2 = 330.

Mathematica

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

Crossrefs

Cf. A002378, A385022, A142463.

Sequence in context: A003742 A324099 A135195 * A273031 A358485 A001509

Adjacent sequences: A385183 A385184 A385185 * A385188 A385189 A385190

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Jun 20 2025

Extensions

Offset and first term corrected by Andrew Howroyd, Nov 12 2025

Status

approved