cp's OEIS Frontend

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

%I A060011 #31 Nov 28 2022 02:38:21
%S A060011 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,
%T A060011 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,
%U A060011 9,9,9,9,9,9,9,9,9,9,9,9,9,2,1,3,4,8,9
%N A060011 Schizophrenic sequence: these are the repeating digits in the decimal expansion of sqrt(f(2n+1)), where f(m) = A014824(m).
%C A060011 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.
%C A060011 From _Peter Bala_, Sep 27 2015: (Start)
%C A060011 Conjecture: same as the repeating digits in the decimal expansion of 1/9*sqrt(1 - 1/10^n).
%C A060011 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)
%C A060011 Conjecture: same as the digital root of the trisection of the Catalan numbers: a(n) = A130856(3*n). - _Christian Krause_, Nov 26 2022
%D A060011 J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 29-36. ASIN: B002ACVZ6O [From _Jason Earls_, Nov 22 2009]
%D A060011 C. A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001. p. 210-211.
%H A060011 K. S. Brown, <a href="http://www.mathpages.com/home/kmath404.htm">Mock-rational numbers</a>.
%H A060011 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a060/A060011.java">Java program</a> (github)
%H A060011 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>
%F A060011 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.
%e A060011 From _Peter Bala_, Sep 27 2015: (Start)
%e A060011 Decimal expansion of 1/9*sqrt(1 - 1/10^20) with repeating strings of digits shown in parentheses for clarity:
%e A060011 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....
%e A060011 Repeating digits 1, 5, 6, 2, 4, 9, 6, 3, 9, 2, 1, 3. (End)
%Y A060011 Cf. A014824.
%K A060011 nonn,base
%O A060011 0,2
%A A060011 _Jason Earls_, Mar 15 2001
%E A060011 Corrected by _Martin Renner_, Apr 15 2007
%E A060011 More terms from _Jinyuan Wang_, Oct 11 2020