A081780 Continued cotangent for square root of 3.

1, 3, 16, 405, 672320, 1352290629546, 4817801789405093582066881, 125242334906904794515426191021121381708761081802849, 48098205042808280233282792304806428848378744166690329294696649253234314988369597043479633480454760546

Offset: 0

Author

Benoit Cloitre, Apr 10 2003

Keywords

• nonn

Crossrefs

Cf. A002666, A002667.

Programs

Mathematica

Floor[NestList[(#*Floor[#]+1)/(#-Floor[#]) &, Sqrt[3], 8]] (* Stefano Spezia, Apr 24 2025 *)

PARI

\p500
bn=vector(100); bn[1]=sqrt(3); b(n)=if(n<0,0,bn[n]);
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(3) = cot(Sum_{n>=0} (-1)^n*acot(a(n))).

Let b(0) = sqrt(3), b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))) then a(n) = floor(b(n)).