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A188934
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Decimal expansion of (1+sqrt(17))/4.
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4
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1, 2, 8, 0, 7, 7, 6, 4, 0, 6, 4, 0, 4, 4, 1, 5, 1, 3, 7, 4, 5, 5, 3, 5, 2, 4, 6, 3, 9, 9, 3, 5, 1, 9, 2, 5, 6, 2, 8, 6, 7, 9, 9, 8, 0, 6, 3, 4, 3, 4, 0, 5, 1, 0, 8, 5, 9, 9, 6, 5, 8, 3, 9, 3, 2, 7, 3, 7, 3, 8, 5, 8, 6, 5, 8, 4, 4, 0, 5, 3, 9, 8, 3, 9, 6, 9, 6, 5, 9, 1, 2, 7, 0, 2, 6, 7, 1, 0, 7, 4, 1, 7, 1, 1, 3, 6, 0, 1, 0, 2, 3, 4, 8, 0, 3, 5, 3, 5, 4, 0, 3, 8, 2, 5, 3, 5, 5, 2, 1, 0
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OFFSET
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1,2
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COMMENTS
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Decimal expansion of the length/width ratio of a (1/2)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.
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LINKS
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EXAMPLE
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1.2807764064044151374553524639935192562...
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MATHEMATICA
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r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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