A042975 -id:A042975 - cp's OEIS frontend

A042974 n 1's followed by a 2.

1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane

The number 1.21121112... is irrational. - Robert G. Wilson v, Jul 05 2000

Fixed point of the following mapping w. Suppose x(n) takes values in {1,2} and w(n)=[x(1),x(2),...,x(n)]. Next define w(n+1)=[x(1),x(2),...,x(n),x(n-1)] if x(n)=1 and w(n+1)=[x(1),x(2),...,x(n),x(n-1),x(n-1)] if x(n)=2. Then taking x(1)=1 and x(2)=2 we get w(infinity)=A042974 (i.e., x(n)=A042974(n)). - Benoit Cloitre, Jan 11 2013

			1.211211121111211111211111121111111211111111211111111121111111111211111...

Harry J. Smith, Table of n, a(n) for n = 1..20000
C. V. Eynden, Problem 10439. An irrational mimic of 1/9, Amer. Math. Monthly, 104 (1997), 873.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A042975, A056030 (continued fraction).

a042974 n = a042974_list !! (n-1)
a042974_list =  1 : 2 :
   concat (zipWith replicate (tail a042974_list) a042974_list)
-- Reinhard Zumkeller, Dec 08 2011

Table[PadLeft[{2},n,1],{n,2,20}]//Flatten (* Harvey P. Dale, Sep 11 2019 *)

{a(n) = 1 + issquare(9 + 8*n)}; /* Michael Somos, Jan 12 2000 */

{ default(realprecision, 20080); x=0; for (n=1, 20000, x=10*x + 1 + issquare(9+8*n)); x/=10^19999; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b042974.txt", n, " ", d)); } \\ Harry J. Smith, May 08 2009

sum(k=1,sqrtint(8*default(realprecision)+9),10^-(2*k-1)^2, 1/9.) \\ Charles R Greathouse IV, Feb 05 2025

A230517 An irrational x such that the decimal representation of neither x nor sqrt(x) contains the digit 0.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Michel Marcus, Oct 22 2013

The rational number 1/9 is an example of a number in [0, 1] such that the decimal representation of neither x nor sqrt(x) contains the digit 0. The object of Problem 10439 of the Amer. Math. Monthly was to find an irrational with the same property (see link).
The solution proposed by Jerrold Grossman defines a sequence of irrationals starting with c1= 0.121121112... (A042974). Moving from left to right, the 0's in the decimal expansion of sqrt(cn) are eliminated by increasing the corresponding digit in the decimal expansion of cn by 2. The limit of cn is a number with the desired property.
The indices of the decimals that are successively changed are 4, 8, 29, 38, 40, 54, 62, 70, 72, 96, 118, ... (see print(ndeci) in PARI script).
The decimal expansion of sqrt(x) begins with 0.3483118317127931144162557719319698175373163374567....

			0.12132113211112111112111111213111112113131112111111111411111113...

C. V. Eynden, Problem 10439. An irrational mimic of 1/9, Amer. Math. Monthly, 104 (1997), 873.

Cf. A042974, A042975.

pdeci(x, nb) = {x = x * 10; for (n=1, nb, d = floor(x); x = (x-d)*10; print1(d, ", ");); print();}
finddeci(x) = {x = x * 10; found = 0; nd = 1; while (! found, d = floor(x); x = (x-d)*10; if (d == 0, found = 1, nd++);); nd;}
changedeci(x, ndeci) = {deci = floor(x * 10^ndeci) - 10*floor(x * 10^(ndeci-1)); x += 2/10^ndeci; x;}
lista(nn) = {prec = 2*nn; default(realprecision, prec); x = 0; for (n=1, prec, x = 10*x + 1 + issquare(9+8*n);); x /= 10^prec; ok = 0; while (! ok, y = sqrt(x); ndeci = finddeci(y); print1(ndeci, ", "); x = changedeci(x, ndeci); if (ndeci > nn, ok =1);); print(); pdeci(x, nn); print("sqrt(x)=", sqrt(x));} \\ Michel Marcus, Oct 22 2013

cp's OEIS Frontend

A042974 n 1's followed by a 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

A230517 An irrational x such that the decimal representation of neither x nor sqrt(x) contains the digit 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI