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Decimal expansion of sqrt(4+sqrt(15)).
1

%I #8 Feb 20 2024 16:33:54

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

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

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

%N Decimal expansion of sqrt(4+sqrt(15)).

%C Decimal expansion of the length/width ratio of a sqrt(6)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

%C A sqrt(6)-extension rectangle matches the continued fraction [2,1,4,6,1,1,2,25,3,...] for the shape L/W=sqrt(4+sqrt(15)). 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 sqrt(6)-extension rectangle, 2 squares are removed first, then 1 square, then 4 squares, then 6 squares,..., so that the original rectangle of shape sqrt(4+sqrt(15)) is partitioned into an infinite collection of squares.

%F Equals A115754 + 10*A020797. - _Hugo Pfoertner_, Feb 20 2024

%e 2.8058837014757787150980888095693049628427513...

%t r = 6^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]]

%t ContinuedFraction[t, 120]

%o (PARI) sqrt(3/2) + sqrt(5/2) \\ _Hugo Pfoertner_, Feb 20 2024

%Y Cf. A020797, A115754, A188925.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Apr 13 2011