Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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