A060011 Schizophrenic sequence: these are the repeating digits in the decimal expansion of sqrt(f(2n+1)), where f(m) = A014824(m).

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

Offset

0,2

Comments

The repeating strings that form the sequence 1, 5, 6, 2, 4, 9, 6, 3, 9, ... become progressively smaller and the irregular strings increase, until eventually the repeating strings disappear. With larger odd values of n however, the demise of the repeating digits slows down.

From Peter Bala, Sep 27 2015: (Start)

Conjecture: same as the repeating digits in the decimal expansion of 1/9*sqrt(1 - 1/10^n).

As n increases, the decimal expansion of 1/9*sqrt(1 - 1/10^n) begins with long strings of repeating digits of 1's, 5's, 6's, 2's,..., which appear to be taken from an initial subsequence of the present sequence, interlaced with the digit strings [0, 41, 597, 178819, 140624, 77213541, 487630208, 1878662109374, 87877739800347, 1191830105251736, 02212270100911458, ...]. An example is given below. Empirical observations: for a fixed value of n, the lengths of the repeating strings gradually shorten until they eventually disappear; as n increases, the number of repeating strings of digits increases. (End)

Conjecture: same as the digital root of the trisection of the Catalan numbers: a(n) = A130856(3*n). - Christian Krause, Nov 26 2022

References

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 29-36. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]

C. A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001. p. 210-211.

Links

Table of n, a(n) for n=0..86.

K. S. Brown, Mock-rational numbers.

Sean A. Irvine, Java program (github)

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Formula

sqrt(f(n)) where f(n) = 10 * f(n-1) + n, for odd integers n. 1, 5, 6, 2, 4, 9, 6, 3, 9, 2, ... are the repeating digits that alternate with random looking strings.

Example

From Peter Bala, Sep 27 2015: (Start)
Decimal expansion of 1/9*sqrt(1 - 1/10^20) with repeating strings of digits shown in parentheses for clarity:
0.(111...111)0(555...555)41(666...666)597(222...222)178819(444...444)140624(999...999)77213541(666...666)487630208(333...333)1878662109374(999...999)87877739800347(222222)1191830105251736(1111)02212270100911458(333)2....
Repeating digits 1, 5, 6, 2, 4, 9, 6, 3, 9, 2, 1, 3. (End)

Crossrefs

Cf. A014824.

Sequence in context: A354214 A210522 A211394 * A021068 A286300 A091873

Adjacent sequences: A060008 A060009 A060010 * A060012 A060013 A060014

Keyword

nonn,base

Author

Jason Earls, Mar 15 2001

Extensions

Corrected by Martin Renner, Apr 15 2007

More terms from Jinyuan Wang, Oct 11 2020

Status

approved