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