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%I #8 Aug 03 2020 02:33:26
%S 2,5,2,7,5,2,5,2,3,1,6,5,1,9,4,6,6,6,8,8,6,2,6,8,2,3,9,7,9,0,9,3,3,6,
%T 1,6,2,9,9,4,8,1,8,8,5,8,9,2,2,6,5,7,3,0,0,8,6,9,0,8,0,7,0,7,9,6,8,9,
%U 5,6,1,4,1,8,4,9,2,5,6,9,6,2,2,0,1,4,5,3,8,5,3,1,6,4,4,8,1,6,7,7,5,5,9,2,0,0,3,0,1,7,9,9,1,9,5,2,4,6,9,5
%N Decimal expansion of (3+sqrt(21))/3.
%C The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/3, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [2, 3/2, 2, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [2,1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.
%F Equals 1 + Sum_{k>=0} binomial(2*k,k)/7^k. - _Amiram Eldar_, Aug 03 2020
%e 2.527525231651946668862682397909336162995...
%t FromContinuedFraction[{2, 3/2, {2, 3/2}}]
%t ContinuedFraction[%, 100] (* [2,1,1,8,1,1,2,... *)
%t RealDigits[N[%%, 120]] (* A190290 *)
%t N[%%%, 40]
%Y Cf. A188635, A190289.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, May 07 2011