A188636 Decimal expansion of length/width of a metasilver rectangle.

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

Offset: 1

Author

Clark Kimberling, Apr 06 2011

Keywords

• nonn
• cons
• easy

Comments

A metasilver rectangle is introduced here as a rectangle such that if a silver rectangle is removed from one end, the remaining rectangle is metasilver. Recall that a rectangle is silver if the removal of 2 squares from one end leaves a rectangle having the same shape s=(length/width) as the original. This metasilver ratio is given by

s=2.774622899504489263198249637919477554666...;

s=[r,r,r,r...], a periodic continued fraction, r=1+sqrt(2);

s=[2,1,3,2,3,2,7,1,1,114,11,1,2,1,...], as at A188637.

Links

• Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Crossrefs

Cf. A188637, A136319, A188635, A188638, A188639.

Programs

Mathematica

t=1+2^(1/2); r=(t+(t^2+4)^(1/2))/2
FullSimplify[r]
N[r, 130]
RealDigits[N[r, 130]][[1]]

Formula

Equals (1+sqrt(2)+sqrt(H))/2, where H=7+2*sqrt(2).