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A188636
Decimal expansion of length/width of a metasilver rectangle.
3
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
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.
FORMULA
Equals (1+sqrt(2)+sqrt(H))/2, 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
KEYWORD
nonn,cons,easy
AUTHOR
Clark Kimberling, Apr 06 2011
STATUS
approved