cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A139030 Real part of (4 + 3i)^n.

Original entry on oeis.org

1, 4, 7, -44, -527, -3116, -11753, -16124, 164833, 1721764, 9653287, 34182196, 32125393, -597551756, -5583548873, -29729597084, -98248054847, -42744511676, 2114245277767, 17982575014036, 91004468168113, 278471369994004, -47340744250793, -7340510203856444, -57540563024581727
Offset: 0

Views

Author

Gary W. Adamson, Apr 06 2008

Keywords

Comments

sqrt (a(n)^2 + (A139031(n))^2) = 5^n. Example: a(3) = -44, A139031(3) = 117. Sqrt (-44^2 + 117^2) = 5^3.
If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 11 divides a(6n+3) for n >= 0; 31 divides a(8n+4) for n>= 0. See the Renault paper in Links. - Clark Kimberling, Oct 02 2024

Examples

			a(5) = -3116 since (4 + 3i)^5 = (-3116 - 237i) where -237 = A139031(5).
		

Crossrefs

Programs

  • Maple
    a:= n-> Re((4+3*I)^n):
    seq(a(n), n=0..24);  # Alois P. Heinz, Oct 15 2024
  • Mathematica
    Re[(4+3I)^Range[40]] (* or *) LinearRecurrence[{8,-25},{4,7},40] (* Harvey P. Dale, Nov 09 2011 *)
    A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
    {a, b, c} = {3, 4, 5};
    Table[TrigExpand[5^n Cos[n (A[b, c, a] - A[c, a, b])]], {n, 0, 50}] (* Clark Kimberling, Oct 02 2024 *)

Formula

Real part of (4 + 3i)^n. Term (1,1) of [4,-3; 3,4]^n. a(n), n>=2 = 8*a(n-1) - 25*a(n-2), given a(0) = 1, a(1) = 4. Odd-indexed terms of A066770 interleaved with even-indexed terms of A066771, irrespective of sign.
G.f.: (1-4*x) / ( 1-8*x+25*x^2 ). - R. J. Mathar, Feb 05 2011
a(n) = 5^n * cos(nB-nC), where B is the angle opposite side CA and C is the angle opposite side AB in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle. - Clark Kimberling, Oct 02 2024
E.g.f.: exp(4*x)*cos(3*x). - Stefano Spezia, Oct 03 2024

Extensions

More terms from Harvey P. Dale, Nov 09 2011
a(0)=1 prepended by Alois P. Heinz, Oct 15 2024

A376283 a(n) = (40)^n * cos(nB), where B is the angle opposite side CA 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, 24, -448, -59904, -2158592, -7766016, 3080978432, 160312590336, 2765438844928, -123759079981056, -10365137990975488, -299512095597133824, 2207640196898357248, 585186082406535266304, 24556707640476321185792, 242424234892406990831616
Offset: 0

Views

Author

Clark Kimberling, Oct 02 2024

Keywords

Comments

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(4n+2) for n >= 0; 17 divides a(8n+4) for n>= 0. See the Renault paper in References.

References

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

Crossrefs

Programs

  • Mathematica
    (*Program 1*)
    A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
    Table[TrigExpand[(20)^n  Cos[n  A[4, 5, 3]]], {n, 0, 30}]
    (*Program 2*)
    LinearRecurrence[{48, -1600}, {1, 24}, 30]

Formula

a(n) = (40)^n * cos(nB), where B is the angle opposite side CA in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.
a(n) = 48 a(n-1) - 1600 a(n-2), where a(0) = 1, a(1) = 24.
From Stefano Spezia, Oct 03 2024: (Start)
G.f.: (1 - 24*x)/(1 - 48*x + 1600*x^2).
E.g.f.: exp(24*x)*cos(32*x). (End)

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

Views

Author

Clark Kimberling, Oct 03 2024

Keywords

Comments

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.

References

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

Crossrefs

Programs

  • Mathematica
    (*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]

Formula

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)
Showing 1-3 of 3 results.