A051382 Numbers k whose base 3 expansion matches (0|1)*(02)?(0|1)* (no more than one "02" allowed in midst of 0's and 1's).

0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 18, 19, 21, 22, 27, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 40, 54, 55, 57, 58, 63, 64, 66, 67, 81, 82, 83, 84, 85, 87, 88, 90, 91, 92, 93, 94, 99, 100, 102, 103, 108, 109, 110, 111, 112, 114, 115, 117, 118, 119, 120, 121, 162, 163

Offset

1,3

Comments

Representation of 2n in base 3 consists entirely of 0's and 2's, except possibly for a single pair of adjacent 1's among them.

9 divides neither C(2s-1,s) [= A001700(s)] nor C(2s,s) [= A000984(s)] if and only if s = a(n). [Cf. also A249721].

Links

Alois P. Heinz, Table of n, a(n) for n = 1..32768 (first 8193 terms from Antti Karttunen)

Eric Weisstein's World of Mathematics, Binomial Coefficient

Index entries for 3-automatic sequences.

Example

In base 3 the terms look like 0, 1, 2, 10, 11, 20, 21, 100, 101, 102, 110, 111, 200, 201, 210, 211, 1000, 1001, 1002, 1010, 1011, 1020, 1021, 1100, 1101, 1102, 1110, 1111, 2000, 2001, 2010, 2011, 2100, 2101, 2 110, 2111, 10000

Maple

q:= n-> (l-> (h-> h=0 or h=1 and l[1+ListTools[Search](2, l)]
          =0 )(numboccur(l, 2)))([convert(n, base, 3)[], 0]):
select(q, [$0..163])[];  # Alois P. Heinz, Jun 28 2021

Prog

(Perl) sub conv_x_base_n { my($x, $b) = @_; my ($r, $z) = (0, ''); do { $r = $x % $b; $x = ($x - $r)/$b; $z = "$r" . $z; } while(0 != $x); return($z); }
(Perl) for($i=1; $i <= 201; $i++) { if(("0" . conv_x_base_n($i, 3)) =~ /^(0|1)*(02)?(0|1)*$/) { print $i, ", "; } }
(Scheme, with Antti Karttunen's IntSeq-library)
(define A051382 (MATCHING-POS 0 0 in_A051382?))
(define (in_A051382? n) (let loop ((n n) (seen02yet? #f)) (cond ((zero? n) #t) ((= 1 n) #t) ((modulo n 3) => (lambda (r) (cond ((= r 2) (if (or seen02yet? (not (zero? (modulo (/ (- n r) 3) 3)))) #f (loop (/ (- n r) 3) #t))) (else (loop (/ (- n r) 3) seen02yet?))))))))
(Python)
import re
from sympy.ntheory.digits import digits
def b3(n): return "".join(map(str, digits(n, 3)[1:]))
def ok(n): return re.fullmatch('2(0|1)*|(0|1)*(02)?(0|1)*', b3(n)) != None
print(list(filter(ok, range(164)))) # Michael S. Branicky, Jun 26 2021

Crossrefs

Complement: A249719.

Terms of A249721 halved.

Cf. A046097, A048645, A037468, A005836, A117966, A249720.

Sequence in context: A065904 A039108 A020756 * A026514 A285974 A227194

Adjacent sequences: A051379 A051380 A051381 * A051383 A051384 A051385

Keyword

nonn,base,easy

Author

David W. Wilson, Antti Karttunen, Oct 24 1999

Extensions

a(0) = 0 prepended as a border-line case by Antti Karttunen, Nov 14 2014

Offset changed to 1 by Georg Fischer, Jun 28 2021

Status

approved