A188638 -id:A188638 - cp's OEIS frontend

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

Clark Kimberling, Apr 06 2011

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.

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

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

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

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

A188639 Decimal expansion of length/width of a 2nd electrum rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2011

See A188619 for the definition of 2nd electrum rectangle. Briefly, it is a rectangle such that the removal of a silver rectangle from one end leaves a golden rectangle.

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

Cf. A188619, A188618, A188638.

t=1+3^(1/2); r=(t+(t^2+4)^(1/2))/2
FullSimplify[r]
N[r, 130]

cp's OEIS Frontend

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

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