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%I #20 Jun 28 2023 08:35:28
%S 1,6,8,68,88,688,888,6888,8888,68888,88888,688888,888888,6888888,
%T 8888888,68888888,88888888,688888888,888888888,6888888888,8888888888,
%U 68888888888,88888888888,688888888888,888888888888,6888888888888,8888888888888,68888888888888
%N Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).
%C Equivalently, with offset 0, least positive integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A249572 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.
%C For n > 2, a(n) + a(n+1) divides the plane into 2 regions. For n > 1, a(2n) - a(2n-1) divides the plane into n+1 regions. For n >= 1, a(2n+1) - a(2n) divides the plane into n regions. - _Ivan N. Ianakiev_, Feb 23 2015
%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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 10, -10).
%F a(n) = 10*a(n-2) + 8 for n >= 4.
%F From _Chai Wah Wu_, Jul 12 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 - 8*x^2 + 5*x + 1)/((x - 1)*(10*x^2 - 1)). (End)
%e The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller positive integer does this, so a(4) = 68.
%t Join[{1, 6, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* _Vincenzo Librandi_, Nov 16 2014 *)
%o (Magma) I:=[1,6,8,68]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2014
%Y Cf. A249572, A250257, A250258, A001743, A001744, A001745, A001746, A002282.
%K nonn,base,easy
%O 1,2
%A _Rick L. Shepherd_, Nov 15 2014
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