A280136 -id:A280136 - cp's OEIS frontend

A280135 Negative continued fraction of Pi (also called negative continued fraction expansion of Pi).

4, 2, 2, 2, 2, 2, 2, 17, 294, 3, 4, 5, 16, 2, 3, 4, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Randy L. Ekl, Dec 26 2016

nonn

Appears that these terms are related to continued fraction of Pi through simple transforms; original continued fraction terms X,1 -> negative continued fraction term X+2 (e.g., 15,1->17, and 292,1->294); other transforms are to be determined.

			Pi = 4 - (1 / (2 - (1 / (2 - (1 / ...))))).

Leonard Eugene Dickson, History of the Theory of Numbers, page 379.

Cf. A001203 (continued fraction of Pi).
Cf. A133593 (exact continued fraction of Pi).
Cf. A280136 (negative continued fraction of e).

\p10000; p=Pi;for(i=1,300,print(i," ",ceil(p)); p=ceil(p)-p;p=1/p )

A286016 Signed continued fraction expansion with all signs negative of tanh(1).

Original entry on oeis.org

1, 5, 2, 2, 2, 2, 9, 2, 2, 2, 2, 2, 2, 2, 2, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 17, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 21, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Kutlwano Loeto, Apr 30 2017

For any given sequence of signs (e_1, e_2, ..., e_n, ...) one may define the signed continued fraction expansion of a real number x by using floor or ceiling in the step i according to e_i = +1 or e_i = -1. In the present case for the sequence (-1, -1, -1, -1, ...) consisting of only negative signs the ceiling is taken at each step, and the formulas with x_0 = x are a_n = ceiling(x_n) and x_{n+1} = 1/(a_n - x_n). See chapter 1 and 2 of the book by Perron.

			a(2) = 5, a(3) = a(4) = a(5) = a(6) = 2, a(7) = 9, etc. These numbers are obtained from the partial quotients xj as follows:
x2 =  (1 +  e^2)/( 2 + 0e^2) ~4.17 so that a(2)=ceiling(x2)=5;
x3 =  (2 + 0e^2)/( 9 - e^2)  ~1.21 so that a(3)=ceiling(x3)=2;
x4 =  (9 -  e^2)/(16 - 2e^2) ~1.31 so that a(4)=ceiling(x4)=2;
x5 = (16 - 2e^2)/(23 - 3e^2) ~1.46 so that a(5)=ceiling(x5)=2;
x6 = (23 - 3e^2)/(30 - 4e^2) ~1.87 so that a(6)=ceiling(x6)=2;
x7 = (30 - 4e^2)/(37 - 5e^2) ~8.11 so that a(7)=ceiling(x7)=9.
The pairs of integers appearing in the xj's are obtained as the principal or as every other of the non-principal approximating fractions of e^2 in the sense of the A. Hurwitz reference.

Adolf Hurwitz, Über die Kettenbrüche, deren Teilnenner arithmetische Reihen bilden, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Jahrg. 41 (1896), Jubelband 2, 34-64. Reprinted in Hurwitz's Mathematische Werke.
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1930.

Cf. A004273 (continued fraction of tanh(1)), A280135, A280136.

x:=(exp(1)-exp(-1))/(exp(1)+exp(-1)):b:=ceil(x): x1:=1/(b-x):L:=[b]:
for k from 0 to 40 do:
b1:=ceil(x1): x1:=1/(b1-x1): L:=[op(L),b1]: od: print(L);

Using an obvious condensed notation we get for the sequence 1, 5, 2^(4), 9, 2^(8), 13, 2^(12), 17, 2^(16), 21, 2^(20), ... where 2^(m) means m copies of 2.

More terms from Jinyuan Wang, Jul 02 2022

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A280135 Negative continued fraction of Pi (also called negative continued fraction expansion of Pi).

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A286016 Signed continued fraction expansion with all signs negative of tanh(1).

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