A066771 -id:A066771 - cp's OEIS frontend

A139031 Imaginary part of (4 + 3i)^n.

3, 24, 117, 336, -237, -10296, -76443, -354144, -922077, 1476984, 34867797, 242017776, 1064447283, 2465133864, -6890111163, -116749235904, -761741108157, -3175197967656, -6358056037323, 28515500892816, 387075408075603, 2383715742284424, 9392840736385317, 15549832333971936

Offset: 1

Gary W. Adamson, Apr 06 2008

Division of each term by 3 generates an integer sequence 1, 8, 39, 112, -79, -3432, -25481, -118048, -307359, 492328, ... - R. J. Mathar, Apr 08 2008

			a(3) = 117 since (4 + 3i)^3 = (-44 + 117i).
a(4) = 336 = 8*a(3) - 25*a(2) = 8*117 - 25*24.
a(3) = 117 = term (2,1) of [4,-3; 3,4]^3.

Index entries for linear recurrences with constant coefficients, signature (8,-25).

Cf. A139030, A066771, A066776.

a[n_]:=Im[(4+3I)^n];Array[a,24] (* James C. McMahon, Jun 25 2025 *)

a(n) = imag((4 + 3*I)^n); \\ Michel Marcus, Jun 25 2025

Imaginary part of (4 + 3i)^n.
Term (2,1) of [4,-3; 3,4]^n.
a(n)^2 + A139030(n)^2 = 5^(2*n).
a(n) = 8*a(n-1) - 25*a(n-2), n>2, given a(1) = 3, a(2) = 24.
(unsigned): Odd-indexed terms of A066771 interleaved with even-indexed terms of A066776.
O.g.f.: 3*x/(1-8*x+25*x^2). - R. J. Mathar, Apr 08 2008

A376285 a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

Original entry on oeis.org

1, 16, 112, -2816, -134912, -3190784, -48140288, -264175616, 10802495488, 451350102016, 10122205069312, 143370521411584, 538974657445888, -40101019526365184, -1498822487822041088, -31921911799759241216, -421972182463479283712, -734345118927640592384

Offset: 0

Clark Kimberling, Oct 03 2024

If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 7 divides a(4*n+2) for n >= 0; 17 divides a(8*n+3) for n >= 0. See the Renault paper in References.

Marc Renault, "The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m", Math. Mag. 86 (2013) 372 - 380.

Index entries for linear recurrences with constant coefficients, signature (32,-400).

Cf. A066771, A374880, A139030, A376455.

(*Program 1*)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
Table[TrigExpand[(20)^n  Cos[n  A[3, 4, 5]]], {n, 0, 30}]
(*Program 2*)
LinearRecurrence[{32, -400}, {1, 16}, 30]

a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.
a(n) = 32*a(n-1) - 400*a(n-2), where a(0) = 1, a(1) = 16.
From Stefano Spezia, Oct 03 2024: (Start)
G.f.: (1 - 16*x)/(1 - 32*x + 400*x^2).
E.g.f.: exp(16*x)*cos(12*x). (End)

A105667 1/(2k)-th of area of primitive Pythagorean triangle with hypotenuse 5^(2^n), where k is the product of all Mersenne primes not exceeding 2^(n+2) - 1.

Original entry on oeis.org

1, 2, 4216, 44834576

Offset: 0

Lekraj Beedassy, May 04 2005

F. Barnes, Divisors of primitive Pythagorean triangles when the hypotenuse is to a power

Cf. A066770; A066771; A098918.

cp's OEIS Frontend

A139031 Imaginary part of (4 + 3i)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376285 a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A105667 1/(2k)-th of area of primitive Pythagorean triangle with hypotenuse 5^(2^n), where k is the product of all Mersenne primes not exceeding 2^(n+2) - 1.

Views

Author

Keywords

Links

Crossrefs