A189040 -id:A189040 - cp's OEIS frontend

A189041 Continued fraction for (e+sqrt(-4+e^2))/2.

2, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 3, 2, 1, 1, 1, 2, 23, 1, 1, 2, 33, 1, 1, 3, 1, 1, 20, 5, 40, 1, 3, 1, 25, 1, 2, 1, 26, 1, 6, 3, 1, 5, 5, 7, 1, 24, 5, 14, 1, 1, 2, 1, 8, 2, 1, 1, 10, 1, 1, 2, 1, 2, 2, 1440, 2, 17, 1, 1, 3, 4, 1, 18, 2, 1, 1, 1, 8, 1, 5, 1, 4, 18, 1, 3, 5, 1, 24, 3, 4, 1, 1, 4, 4, 1, 137, 3, 1, 1, 1, 6, 1, 1, 7, 1, 1, 2, 6, 1, 1, 5, 1, 2, 5, 7, 1, 3

Offset: 0

Clark Kimberling, Apr 15 2011

			(e+sqrt(-4+e^2))/2=[2,3,1,1,2,1,3,2,1,1,2,2,...]

Cf. A189040 (decimal expansion).

r = E; t = (r + (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Offset changed by Andrew Howroyd, Jul 08 2024

A189042 Decimal expansion of (e-sqrt(-4+e^2))/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 15 2011

Decimal expansion of the shape (= length/width = (e-sqrt(-4+e^2))/2) of the lesser e-contraction rectangle.

See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.

			0.4386714553485326087582705944364891354570...

Cf. A188738, A189040, A189041.

r = E; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

(exp(1)-sqrt(exp(2)-4))/2 \\ Charles R Greathouse IV, Apr 25 2016

a(129) corrected by Georg Fischer, Apr 04 2020

A258898 a(1)=a(2)=1, a(n) = ceiling(e*a(n-1) - a(n-2)) for n>2.

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 65, 149, 341, 778, 1774, 4045, 9222, 21023, 47925, 109251, 249051, 567740, 1294227, 2950334, 6725613, 15331778, 34950481, 79673480, 181624492, 414033077, 943834098, 2151574001, 4904750412, 11180919918

Offset: 1

Morris Neene, Jun 14 2015

nonn

Ratio of consecutive terms approaches A189040, (e + sqrt(e^2 - 4))/2.

			a(2) = ceiling(e*1 - 1) = 2;
a(3) = ceiling(e*2 - 1) = 5;
a(4) = ceiling(e*5 - 2) = 12;
a(5) = ceiling(e*12 - 5) = 28.

Cf. A001113, A189040.

I:=[1,1]; [n le 2 select I[n] else Ceiling(Exp(1)*Self(n-1)-Self(n-2)): n in [1..200]];

a:= proc(n) option remember; `if`(n<3, 1,
      ceil(exp(1)*a(n-1)-a(n-2)))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 18 2015

nxt[{a_,b_}]:={b,Ceiling[E*b-a]}; NestList[nxt,{1,1},30][[All,1]] (* Harvey P. Dale, Dec 02 2017 *)

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A189041 Continued fraction for (e+sqrt(-4+e^2))/2.

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A189042 Decimal expansion of (e-sqrt(-4+e^2))/2.

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A258898 a(1)=a(2)=1, a(n) = ceiling(e*a(n-1) - a(n-2)) for n>2.

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