A220812 -id:A220812 - cp's OEIS frontend

A051840 Integer part of the Verhulst sequence x(n)=x(n-1)+3(1-x(n-1))x(n-1), x(0)=.1.

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 13 1999

a(n) = floor(A220811(n)/A220812(n)). - Reinhard Zumkeller, Dec 22 2012

Peitgen, H.-O. and Richter, P. H., The Beauty of Fractals. Springer, 1986, p. 24.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Logistic Equation
Wikipedia, Logistic map

a051840 n = a051840_list !! n
a051840_list = map floor vs where
   vs = iterate (\x -> x * (4 - 3 * x)) 0.1
-- Reinhard Zumkeller, Dec 22 2012

NestList[ #*(4-3*#)&, 0.1, 103] // Floor (* Jean-François Alcover, Jan 16 2013 *)

Data corrected for n > 37 by Reinhard Zumkeller, Dec 22 2012

A220811 Numerators of the Verhulst sequence x(n+1)=4x(n)-3x(n)^2, x(0)=1/10.

Original entry on oeis.org

1, 37, 10693, 84699253, 12357810823725973, 36165967884042486031784215609813, 10542455454648216809579521558960739092388747823498758873614315093

Offset: 0

Reinhard Zumkeller, Dec 22 2012

H. O. Peitgen and P. H. Richter, The Beauty of Fractals. Springer, 1986, p. 23f.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9
Eric Weisstein's World of Mathematics, Logistic Equation
Wikipedia, Logistic map

Cf. A051840, A220812 (denominators).

import Data.Ratio ((%), numerator)
a220811 n = a220811_list !! n
a220811_list = map numerator vs where
   vs = iterate (\x -> x * (4 - 3 * x)) (1 % 10)
-- Reinhard Zumkeller, Dec 22 2012

A225162 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 9, 91, 9181, 92480761, 9304615055139121, 93529710772930377727152664652641, 9394835719974970982728198049552322910011762062750179997188274881

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence of fractions f(n) is A165428(n+1), hence sum(A165428(i+1)/a(i),i=1..n) = product(A165428(i+1)/a(i),i=1..n) = A165428(n+2)/A225169(n) = A220812(n-1)/A225169(n).

			f(n) = 10, 10/9, 100/91, 10000/9181, ...
10 + 10/9 = 10 * 10/9 = 100/9; 10 + 10/9 + 100/91 = 10 * 10/9 * 100/91 = 10000/819; ...

Cf. A100441, A165428, A220812, A225169.

b:=n->10^(2^(n-2)); # n > 1
b(1):=10;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(a(i),i=1..8);

a(n) = 10^(2^(n-2)) - product(a(i),i=1..n-1), n > 1 and a(1) = 1.

a(n) = 10^(2^(n-2)) - p(n) with a(1) = 1 and p(n) = p(n-1)*a(n-1) with p(1) = 1.

A225169 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 9, 819, 7519239, 695384944860879, 6470289227069622272847335347359, 605164280025029017271801950447677089988237937249820002811725119

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165428(n+2), hence sum(A165428(i+1)/A225162(i),i=1..n) = product(A165428(i+1)/A225162(i),i=1..n) = A165428(n+2)/a(n) = A220812(n-1)/a(n).

			f(n) = 10, 10/9, 100/91, 10000/9181, ...
10 + 10/9 = 10 * 10/9 = 100/9; 10 + 10/9 + 100/91 = 10 * 10/9 * 100/91 = 10000/819; ...
s(n) = 1/b(n) = 10, 100/9, 10000/819, ...

Cf. A076628, A165428, A220812, A225162.

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:
b(1):=1/10;
a:=n->10^(2^(n-1))*b(n);
seq(a(i),i=1..7);

a(n) = 10^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/10.

cp's OEIS Frontend

A051840 Integer part of the Verhulst sequence x(n)=x(n-1)+3*(1-x(n-1))*x(n-1), x(0)=.1.

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A220811 Numerators of the Verhulst sequence x(n+1)=4*x(n)-3*x(n)^2, x(0)=1/10.

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A225162 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A225169 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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Maple

Formula

A051840 Integer part of the Verhulst sequence x(n)=x(n-1)+3(1-x(n-1))x(n-1), x(0)=.1.

A220811 Numerators of the Verhulst sequence x(n+1)=4x(n)-3x(n)^2, x(0)=1/10.