A190288 -id:A190288 - cp's OEIS frontend

A190287 Decimal expansion of (5+sqrt(25+4r))/2, where r=sqrt(5).

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (5+sqrt(25+4r))/2, where r=sqrt(5), can be partitioned into rectangles of shapes 5 and r in a manner that matches the periodic continued fraction [5, r, 5, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [5,2,2,2,1,1,1,10,1,1,2,1,...] at A190288. For details, see A188635.

			5.413085645411028710287065567557494135316...

Cf. A188635, A190288.

r=5^(1/2)
FromContinuedFraction[{5, r, {5, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190288 *)
RealDigits[N[%%, 120]]     (* A190287 *)
N[%%%, 40]

cp's OEIS Frontend

A190287 Decimal expansion of (5+sqrt(25+4r))/2, where r=sqrt(5).

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