A381006 -id:A381006 - cp's OEIS frontend

A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

7, 175, 3007, 48895, 785407, 12578815, 201310207, 3221159935, 51539345407, 824632672255, 13194135339007, 211106215755775, 3377699653419007, 54043195260010495, 864691127381393407, 13835058050987196415, 221360928867334750207, 3541774862083514433535, 56668397794160864657407

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020884.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A020884.

Cf. A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

[2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1: n in [1..20]];

A381005[n_] := (3*# + 2)*(# - 2)/4 & [4^n]; Array[A381005, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {7, 175, 3007}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = 2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1

def A381005(n): return ((m:=1<<(n<<1)-1)-1)*(3*m+1) # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) - 2^(4n-2) - 2^(2n) - 1.
a(n) = sqrt( A381007(n)^2 - A381006(n)^2 ).
G.f.: (7 + 28*x - 80*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381007 Ordered hypotenuses of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

25, 337, 5185, 82177, 1311745, 20975617, 335560705, 5368774657, 85899608065, 1374390583297, 21990236749825, 351843737665537, 5629499601321985, 90071992815845377, 1441151881832300545, 23058430096431906817, 368934881491370901505, 5902958103655775993857

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020882.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A020882.

Cf. A381005 (short legs), A381006 (long legs), A381008 (perimeters), A381009 (areas).

[2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1: n in [1..20]];

A381007[n_] := 5*4^(2*n - 1) + 4^n + 1; Array[A381007, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {25, 337, 5185}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = 2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1

def A381007(n): return (m:=1<<(n<<1)-1)*(5*m+2)+1 # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.
a(n) = sqrt( A381005(n)^2 + A381006(n)^2 ).
G.f.: (25 - 188*x + 208*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

56, 800, 12416, 197120, 3147776, 50339840, 805339136, 12885032960, 206158954496, 3298536980480, 52776566521856, 844424963686400, 13510799016329216, 216172782650654720, 3458764515968024576, 55340232229718589440, 885443715572418215936, 14167099448746374594560

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A024364.

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A024364.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381009 (areas).

[2^(4*n+1) + 2^(2*n+1) + 2^(4*n): n in [1..20]];

A381008[n_] := #*(3*# + 2) & [4^n]; Array[A381008, 20] (* or *)
LinearRecurrence[{20, -64}, {56, 800}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI
```
a(n) = 2^(4*n+1) + 2^(2*n+1) + 2^(4*n)
```

def A381008(n): return (m:=1<<(n<<1))*(2+3*m) # Chai Wah Wu, Feb 13 2025

a(n) = A381005(n) + A381006(n) + A381007(n).
a(n) = 2^(4n+1) + 2^(2n+1) + 2^(4n).
G.f.: 8*(7 - 40*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381009 Ordered areas of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

84, 25200, 6350784, 1614708480, 412583721984, 105570270965760, 27022696873181184, 6917599389942743040, 1770891934572664848384, 453347470584212823736320, 116056897129722086198083584, 29710562123440325102508441600, 7605903676927233379495034486784, 1947111326786263531071061496954880

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A024406.

Paolo Xausa, Table of n, a(n) for n = 1..400
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (340,-22848,348160,-1048576).

Cf. A024406.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters).

[(2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2: n in [1..20]];

A381009[n_] := (3*# + 2)*(# + 2)*(# - 2)*2^(2*n - 3) & [4^n]; Array[A381009, 20] (* or *)
LinearRecurrence[{340, -22848, 348160, -1048576}, {84, 25200, 6350784, 1614708480}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = (2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2

def A381009(n): return (m:=1<<(n<<1)-1)*(m-1)*(m+1)*(3*m+1)<<1 # Chai Wah Wu, Feb 13 2025

a(n) = A381005(n) * A381006(n) / 2.
a(n) = (2^(4n) + 2^(2n+1)) * (2^(4n) - 2^(4n-2) - 2^(2n) - 1) / 2.
G.f.: 12*(7 - 280*x - 24832*x^2 + 163840*x^3)/((1 - 4*x)*(1 - 16*x)*(1 - 64*x)*(1 - 256*x)). - Stefano Spezia, Feb 13 2025

cp's OEIS Frontend

A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381007 Ordered hypotenuses of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381009 Ordered areas of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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Author

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Comments

Links

Crossrefs

Programs

Magma

Mathematica

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Python

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