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Least positive integer whose decimal digits divide the plane into n+1 regions. Equivalently, least positive integer with n holes in its decimal digits.
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%I #64 Sep 01 2023 04:01:24

%S 1,4,8,48,88,488,888,4888,8888,48888,88888,488888,888888,4888888,

%T 8888888,48888888,88888888,488888888,888888888,4888888888,8888888888,

%U 48888888888,88888888888,488888888888,888888888888,4888888888888,8888888888888,48888888888888

%N Least positive integer whose decimal digits divide the plane into n+1 regions. Equivalently, least positive integer with n holes in its decimal digits.

%C Leading zeros are not permitted. Variations are possible depending upon whether 4 is considered "holey" (if not, replace each "4" with a "6") and whether nonnegative integers are permitted (a(2) becomes 0). In each case, all terms after the first could be considered "wholly holey," as could all terms of A001743 and A001744, as each digit contains a hole (loop). The analogous sequence of bits for base 2 is simply A011557, the powers of 10, read instead as binary numbers, i.e., as powers of two.

%H Alois P. Heinz, <a href="/A249572/b249572.txt">Table of n, a(n) for n = 0..2000</a>

%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 Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, <a href="https://arxiv.org/abs/2306.06794">A blind spot for large language models: Supradiegetic linguistic information</a>, arXiv:2306.06794 [cs.CL], 2023.

%H <a href="/index/Ho#holes">Index entries for sequences related to holes in digits</a>

%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 >= 3.

%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.: (10*x^3 - 6*x^2 + 3*x + 1)/((x - 1)*(10*x^2 - 1)). (End)

%e From _Jon E. Schoenfield_, Nov 15 2014: (Start)

%e This sequence uses "holey" fours. So a(1)=4, because

%e . . . . . . . . . . . . . . . . . . . . . . . .

%e . . . .

%e . XXXX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XX XX . . XX XX .

%e . XXXXXXXXXXXXX . . XXXXXXXXXXXXX .

%e . XX . . XX .

%e . XX . . XX .

%e . XX . . XX .

%e . XX . . XX .

%e . XX . . XX .

%e . . . .

%e . "Holey" 4 . . "Non-holey" 4 .

%e . . . . . . . . . . . . . . . . . . . . . . . . (End)

%p a:= n-> `if`(n=0, 1, parse(cat(4*(irem(n, 2, 'q')), 8$q))):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 01 2014

%t LinearRecurrence[{1,10,-10},{1,4,8,48},50]] (* _Paolo Xausa_, May 31 2023 *)

%o (Magma) I:=[1,4,8,48]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // _Vincenzo Librandi_, Nov 17 2014

%o (PARI) A249572(n)=10^(n\2)*if(n%2,45-(n>1)*5,22)\45 \\ "(...,9-(n>1),4.4)\9" would be shorter but cause problems beyond realprecision. - _M. F. Hasler_, Jul 25 2015

%Y Cf. A001743, A001744, A001745, A001746, A002282, A011557, A000079.

%Y The analogous sequence using 6 instead of 4 is A250256. - _N. J. A. Sloane_, Sep 27 2019

%K nonn,base,easy

%O 0,2

%A _Rick L. Shepherd_, Nov 01 2014

%E Offset corrected by Brady Haran, Nov 27 2018