OFFSET
1,2
COMMENTS
Decimal expansion of the length/width ratio of a (1/2)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.
EXAMPLE
1.2807764064044151374553524639935192562...
MATHEMATICA
r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)
PROG
(PARI) (sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Apr 13 2011
STATUS
approved