OFFSET
1,2
COMMENTS
The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+r))/r, where r=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 2 in a manner that matches the periodic continued fraction [r, 2, r, 2, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,4,6,1,2,2,2,1,1 ...] at A190282. For details, see A188635.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
EXAMPLE
1.805790894654357490440645557345527417829...
MATHEMATICA
r=2^(1/2)
FromContinuedFraction[{r, 2, {r, 2}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A190282 *)
RealDigits[N[%%, 120]] (* A190281 *)
N[%%%, 40]
RealDigits[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 31 2018 *)
PROG
(PARI) (1 + sqrt(1 + sqrt(2)))/sqrt(2) \\ G. C. Greubel, Jan 31 2018
(Magma) (1 + Sqrt(1 + Sqrt(2)))/Sqrt(2); // G. C. Greubel, Jan 31 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, May 07 2011
STATUS
approved