cp's OEIS Frontend

A188934 Decimal expansion of (1+sqrt(17))/4.

%I A188934 #19 Aug 25 2025 10:53:25
%S A188934 1,2,8,0,7,7,6,4,0,6,4,0,4,4,1,5,1,3,7,4,5,5,3,5,2,4,6,3,9,9,3,5,1,9,
%T A188934 2,5,6,2,8,6,7,9,9,8,0,6,3,4,3,4,0,5,1,0,8,5,9,9,6,5,8,3,9,3,2,7,3,7,
%U A188934 3,8,5,8,6,5,8,4,4,0,5,3,9,8,3,9,6,9,6,5,9,1,2,7,0,2,6,7,1,0,7,4,1,7,1,1,3,6,0,1,0,2,3,4,8,0,3,5,3,5,4,0,3,8,2,5,3,5,5,2,1,0
%N A188934 Decimal expansion of (1+sqrt(17))/4.
%C A188934 Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
%C A188934 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.
%C A188934 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.
%e A188934 1.2807764064044151374553524639935192562...
%t A188934 r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
%t A188934 (* for the continued fraction *) ContinuedFraction[t, 120]
%t A188934 RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
%t A188934 RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* _Robert G. Wilson v_, Aug 17 2011 *)
%o A188934 (PARI) (sqrt(17)+1)/4 \\ _Charles R Greathouse IV_, May 12 2016
%Y A188934 Cf. A188640, A188935.
%Y A188934 Essentially the same as A188485.
%K A188934 nonn,cons,changed
%O A188934 1,2
%A A188934 _Clark Kimberling_, Apr 13 2011