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

1,1

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

Table of n, a(n) for n=1..130.

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

Formula

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

Mathematica

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

Crossrefs

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

Sequence in context: A164767 A244645 A277527 * A344520 A021977 A057105

Adjacent sequences: A188633 A188634 A188635 * A188637 A188638 A188639

Keyword

nonn,cons

Author

Clark Kimberling, Apr 06 2011

Status

approved