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Least positive integer whose decimal digits divide the plane into n regions (version for people who write 2 with a curlicue).
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%I #31 Jun 27 2023 14:07:03

%S 1,2,8,28,88,288,888,2888,8888,28888,88888,288888,888888,2888888,

%T 8888888,28888888,88888888,288888888,888888888,2888888888,8888888888,

%U 28888888888,88888888888,288888888888,888888888888,2888888888888,8888888888888,28888888888888

%N Least positive integer whose decimal digits divide the plane into n regions (version for people who write 2 with a curlicue).

%C Equivalently, with offset 0, least positive integer with n holes in its decimal digits. Note that 2 written with a curlicue has one hole, 8 has two holes, 28 has three holes, etc., as in the illustration. See A249572 and A250256 for other versions.

%H Brady Haran and N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=OeGSQggDkxI">What Number Comes Next?</a> (2018), Numberphile video

%H N. J. A. Sloane, <a href="/A279193/a279193.jpg">Illustration of initial terms.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 10, -10).

%F From _Chai Wah Wu_, Dec 14 2016: (Start)

%F a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>4.

%F G.f.: x*(10*x^3 - 4*x^2 + x + 1)/((x - 1)*(10*x^2 - 1)). (End)

%F a(n) = ((26 - (13 - 4*sqrt(10))*(1 - (-1)^n))*sqrt(10^n) - 80)/90 for n>1, a(1)=1. - _Bruno Berselli_, Dec 15 2016

%t Join[{1}, LinearRecurrence[{1, 10, -10}, {2, 8, 28}, 30]] (* _Vincenzo Librandi_, Dec 15 2016 *)

%o (Magma) I:=[1,2,8,28]; [n le 4 select I[n] else Self(n-1)+10*Self(n-2)-10*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 15 2016

%Y See A249572, A250256 for other versions.

%K nonn,base,easy

%O 1,2

%A _N. J. A. Sloane_, Dec 14 2016, following a suggestion from Colin Stewart