cp's OEIS Frontend

A188734 Decimal expansion of (7+sqrt(65))/4.

%I A188734 #21 Sep 08 2022 08:45:56
%S A188734 3,7,6,5,5,6,4,4,3,7,0,7,4,6,3,7,4,1,3,0,9,1,6,5,3,3,0,7,5,7,5,9,4,2,
%T A188734 7,8,2,7,8,3,5,9,9,0,7,6,4,0,2,1,4,3,3,4,6,9,8,4,1,4,8,0,9,7,3,1,5,9,
%U A188734 6,8,7,3,7,7,5,6,4,2,2,0,5,0,7,4,0,0,3,8,5,6,6,6,7,9,3,0,7,6,6,0,9,0,9,3,6,0,6,1,6,5,3,4,9,8,6,4,7,8,0,5,3,4,3,7,1,6,3,0,3,0
%N A188734 Decimal expansion of (7+sqrt(65))/4.
%C A188734 Apart from the second digit, the same as A171417. - _R. J. Mathar_, Apr 15 2011
%C A188734 Apart from the first two digits, the same as A188941. - _Joerg Arndt_, Apr 16 2011
%C A188734 Decimal expansion of the length/width ratio of a (7/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
%C A188734 A (7/2)-extension rectangle matches the continued fraction [3,1,3,3,1,3,3,1,3,3,1,3,3,...] for the shape L/W=(7+sqrt(65))/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 (7/2)-extension rectangle, 3 squares are removed first, then 1 square, then 3 squares, then 3 squares,..., so that the original rectangle of shape (7+sqrt(65))/4 is partitioned into an infinite collection of squares.
%H A188734 G. C. Greubel, <a href="/A188734/b188734.txt">Table of n, a(n) for n = 1..10000</a>
%e A188734 3.7655644370746374130916533075759427827835990...
%p A188734 evalf((7+sqrt(65))/4,140); # _Muniru A Asiru_, Nov 01 2018
%t A188734 r = 7/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t A188734 N[t, 130]
%t A188734 RealDigits[N[t, 130]][[1]]
%o A188734 (PARI) default(realprecision, 100); (7+sqrt(65))/4 \\ _G. C. Greubel_, Nov 01 2018
%o A188734 (Magma) SetDefaultRealField(RealField(100)); (7+Sqrt(65))/4; // _G. C. Greubel_, Nov 01 2018
%Y A188734 Cf. A188640.
%K A188734 nonn,cons
%O A188734 1,1
%A A188734 _Clark Kimberling_, Apr 12 2011