A383833 -id:A383833 - cp's OEIS frontend

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

1, 7, 31, 97, 241, 511, 967, 1681, 2737, 4231, 6271, 8977, 12481, 16927, 22471, 29281, 37537, 47431, 59167, 72961, 89041, 107647, 129031, 153457, 181201, 212551, 247807, 287281, 331297, 380191, 434311, 494017, 559681, 631687, 710431, 796321, 889777, 991231, 1101127, 1219921, 1348081

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 11 2025

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000217, A383833, A002061, A006007, A058919, A336535.

a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

a(n) = 2*(A000217(n))^2 + 4*A000217(n) + 1.

a(n) = 6*A006007(n) + 1

A336535 a(n) = (m(n)^2 + 3)(m(n)^2 + 7)/32, where m(n) = 2n - 1.

Original entry on oeis.org

1, 6, 28, 91, 231, 496, 946, 1653, 2701, 4186, 6216, 8911, 12403, 16836, 22366, 29161, 37401, 47278, 58996, 72771, 88831, 107416, 128778, 153181, 180901, 212226, 247456, 286903, 330891, 379756, 433846, 493521, 559153, 631126, 709836, 795691, 889111, 990528, 1100386, 1219141, 1347261, 1485226, 1633528, 1792671

Offset: 1

Jeff Brown, Jul 24 2020

For m(n) = 3,5,11, and 181, the perfect numbers (A000396), 6, 28, 496, and 33550336 are produced, respectively. 3,5,11, and 181 are the numbers m(n) such that (m(n)^2+7) is a power of 2. cf A038198.

The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A000217(n). - Miguel-Ángel Pérez García-Ortega, May 16 2025

			m(2) = 2*2-1 = 3 and (3^2+3)*(3^2+7)/32 = 6, so 6 is in the sequence.

David M. Burton, Elementary Number Theory, McGraw-Hill (2011), 25.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000396, A038198.
Cf. A002061, A014206.
Cf. A000217, A058919, A383833, A383834.

Table[((n^2+3)*(n^2+7))/32, {n,1,100,2}]
a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* Miguel-Ángel Pérez García-Ortega, May 16 2025 *)

a(n)=(1-n+n^2)*(2-n+n^2)/2 \\ Charles R Greathouse IV, Oct 21 2022

From Stefano Spezia, Jul 25 2020: (Start)
O.g.f.: x*(1 + x + 8*x^2 + x^3 + x^4)/(1 - x)^5.
a(n) = (1 - n + n^2)*(2 - n + n^2)/2.
a(n) = A002061(n)*A014206(n-1)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
a(n) = (A000217(n-1)+1)*(2*A000217(n-1)+1). - Miguel-Ángel Pérez García-Ortega, May 16 2025

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

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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A336535 a(n) = (m(n)^2 + 3)(m(n)^2 + 7)/32, where m(n) = 2n - 1.

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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.

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cp's OEIS Frontend

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A336535 a(n) = (m(n)^2 + 3)*(m(n)^2 + 7)/32, where m(n) = 2*n - 1.

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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.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

A336535 a(n) = (m(n)^2 + 3)(m(n)^2 + 7)/32, where m(n) = 2n - 1.