A081781 Continued cotangent for square root of 5.

2, 23, 2961, 9798690, 110528843365255, 16850689692769666225948933906, 357017084742873887104901208145335336566095925224005385073, 137046393407742688281781643318644289249627231296779665713406441509975367310318242383523303176564909084594247359836

Offset: 0

Author

Benoit Cloitre, Apr 10 2003

Keywords

• nonn

Crossrefs

Cf. A002666, A002667.

Programs

Mathematica

Floor[NestList[(#*Floor[#]+1)/(#-Floor[#]) &, Sqrt[5], 7]] (* Stefano Spezia, Apr 24 2025 *)

PARI

bn=vector(100);
b(n)=if(n<0,0,bn[n]);
bn[1]=sqrt(5);
for(n=2,10,bn[n]=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))));
a(n) = floor(b(n+1));

Formula

sqrt(5) = cot(Sum_{n>=0} (-1)^n*acot(a(n))).

Let b(0) = sqrt(5), b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))) then a(n) = floor(b(n)).