A154411 a(n) = period of the sequence [ x(k+1) := floor(2*cos(2*Pi/7)*x(k)) - x(k-1), with x(0) := cosh(n)^2, x(1) := sinh(n)^2 ].

8, 56, 996, 1680, 5236, 280, 28, 224, 448, 749196, 728, 980, 420, 112, 1708, 56, 112, 56, 308, 252, 532, 84, 2156, 28, 84, 280, 29260, 140, 6454336, 3220, 868, 6916, 532, 168, 28, 28, 28, 56, 1568, 28, 56, 376236, 28, 2380, 112, 56, 28, 3640, 84, 84, 84, 168, 224, 56, 4732, 56, 84, 28, 364, 8232

Offset

0,1

Comments

The Pi*(rational) suppresses the natural tendency to grow like x0^2 + y0^2.

All terms of the sequence x considered for a(n) are of the form k +- sinh(n)^2, where k is an integer; in particular, x(0) = x(1) + 1. Are all terms except for a(0) and a(2) a multiple of 28? - M. F. Hasler, Apr 12 2021

Links

M. F. Hasler, Table of n, a(n) for n = 0..130

Example

From Neil Bickford, Apr 12 2021: (Start)
This prints the cycle for a(23)=28:
Block[{x0=Cosh[23]^2,x1=Sinh[23]^2,xPrev,x,k=1},(
  xPrev=x0;
  x=x1;
  Do[
    (Print[{k,xPrev,x}];
    {xPrev,x}={x,Floor[2 Cos[2 Pi/7]x]-xPrev};
    k++),
  29]
)]
This produces:
{1,Cosh[23]^2,Sinh[23]^2}
{2,Sinh[23]^2,29603668079891461525-Cosh[23]^2}
{3,29603668079891461525-Cosh[23]^2,7311502210955022182-Sinh[23]^2}
{4,7311502210955022182-Sinh[23]^2,-50090042030186845236+Cosh[23]^2}
(...several terms omitted...)
{26,-50090042030186845236+Sinh[23]^2,7311502210955022184-Cosh[23]^2}
{27,7311502210955022184-Cosh[23]^2,29603668079891461526-Sinh[23]^2}
{28,29603668079891461526-Sinh[23]^2,Cosh[23]^2}
{29,Cosh[23]^2,Sinh[23]^2}
(End)

Mathematica

(* Mma code from Neil Bickford, Apr 12 2021 *)
func[n_] :=Block[{x0=Cosh[n]^2, x1=Sinh[n]^2, xPrev, x, k=1}, (
  xPrev=x0;
  x=x1;
  While[True,
    ({xPrev, x}={x, Floor[2 Cos[2 Pi/7]x]-xPrev};
    If[x0==xPrev && x1==x, Return[k]];
    k++;
  )]
)]
In[20]:= Table[func[n], {n, 0, 10}]
Out[20]= {8, 56, 996, 1680, 5236, 280, 28, 224, 448, 749196, 728}

Prog

(PARI) A154411(n)={my(a1='s, a0='s+1, s=sinh(n)^2, c=1/2/cos(2*Pi/7), L=Map([a0, [0]; a1, [1]]), t, i); for(n=2, oo, [a0, a1]=[a1, eval(a1)\c-a0]; if( !mapisdefined(L, a1), mapput(L, a1, [n]), i=setintersect(t=mapget(L, a1), [j+1| j<-mapget(L, a0)]), return(n-i[1]), mapput(L, a1, concat(t, n))))} \\ Precision of 115 digits is enough for n <= 130, not beyond. - M. F. Hasler, Apr 12 2021

Crossrefs

Sequence in context: A209072 A133671 A326239 * A105850 A009089 A296466

Adjacent sequences: A154408 A154409 A154410 * A154412 A154413 A154414

Keyword

nonn,more

Author

Bill Gosper, Jun 24 2010

Extensions

Definition corrected by N. J. A. Sloane, Apr 12 2021

More terms from M. F. Hasler, Apr 12 2021

Status

approved