%I #9 Nov 29 2017 03:47:07

%S 1,8,7,0,8,2,8,6,9,3,3,8,6,9,7,0,6,9,2,7,9,1,8,7,4,3,6,6,1,5,8,2,7,4,

%T 6,5,0,8,7,8,0,0,9,9,0,3,8,8,9,3,6,3,4,7,3,1,5,1,8,7,2,7,3,3,6,6,0,0,

%U 1,7,5,7,8,1,5,3,4,6,9,5,1,3,9,8,8,4,0,4,9,4,7,5,9,7,1,8,9,7,8

%N Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

%C The regular continued fraction of sqrt(14)/2 is [1, repeat(1, 6, 1, 2)].

%C The convergents are given in A295336/A295337.

%C sqrt(14)/2 appears in a regular hexagon inscribed in a circle of radius 1 unit in the following way. Draw a straight line through two opposed midpoints of a side (halving the hexagon). The length between one of the midpoints, say M, and one of the two vertices nearest to the opposed midpoint is sqrt(13)/2 = A295330 units. A circle through M with this length ratio sqrt(13)/2 intersects the line below the hexagon at a point, say P. Then the length ratio between P and one of the two vertices nearest to M is sqrt(14)/2 (from a rectangular triangle (1/2, sqrt(13)/2, sqrt(14)/2).

%e 1.87082869338697069279187436615827465087800990388936347315187273366001757815...

%Y Cf. A010471, A295330, A295336/A295337.

%K nonn,cons

%O 1,2

%A _Wolfdieter Lang_, Nov 27 2017