OFFSET
0,2
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.
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 linear recurrences with constant coefficients, signature (1,10,-10).
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)
EXAMPLE
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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Rick L. Shepherd, Nov 01 2014
EXTENSIONS
Offset corrected by Brady Haran, Nov 27 2018
STATUS
approved