A249572 Least positive integer whose decimal digits divide the plane into n+1 regions. Equivalently, least positive integer with n holes in its decimal digits.

1, 4, 8, 48, 88, 488, 888, 4888, 8888, 48888, 88888, 488888, 888888, 4888888, 8888888, 48888888, 88888888, 488888888, 888888888, 4888888888, 8888888888, 48888888888, 88888888888, 488888888888, 888888888888, 4888888888888, 8888888888888, 48888888888888

Offset: 0

Author

Rick L. Shepherd, Nov 01 2014

Keywords

• nonn
• base
• easy

Comments

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.

Examples

			From _Jon E. Schoenfield_, Nov 15 2014: (Start)
This sequence uses "holey" fours. So a(1)=4, because
. . . . . . . . . . . .       . . . . . . . . . . . .
.                     .       .                     .
.           XXXX      .       .    XX       XX      .
.          XX XX      .       .    XX       XX      .
.         XX  XX      .       .    XX       XX      .
.        XX   XX      .       .    XX       XX      .
.       XX    XX      .       .    XX       XX      .
.      XX     XX      .       .    XX       XX      .
.     XX      XX      .       .    XX       XX      .
.    XX       XX      .       .    XX       XX      .
.    XXXXXXXXXXXXX    .       .    XXXXXXXXXXXXX    .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.                     .       .                     .
.      "Holey" 4      .       .    "Non-holey" 4    .
. . . . . . . . . . . .       . . . . . . . . . . . . (End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video.
• Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, A blind spot for large language models: Supradiegetic linguistic information, arXiv:2306.06794 [cs.CL], 2023.
• Index entries for sequences related to holes in digits
• Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Crossrefs

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

The analogous sequence using 6 instead of 4 is A250256. - N. J. A. Sloane, Sep 27 2019

Programs

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

Maple

a:= n-> `if`(n=0, 1, parse(cat(4*(irem(n, 2, 'q')), 8$q))):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2014

Mathematica

LinearRecurrence[{1,10,-10},{1,4,8,48},50] (* Paolo Xausa, May 31 2023 *)

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

Formula

a(n) = 10*a(n-2) + 8 for n >= 3.

From Chai Wah Wu, Dec 14 2016: (Start)

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

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

a(n) = (2/9)*(10^(n/2)*(4*((n+1) mod 2) + 11*sqrt(2/5)*(n mod 2)) - 4) for n >= 1. - Alan Michael Gómez Calderón, May 04 2025

Extensions

Offset corrected by Brady Haran, Nov 27 2018