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

Author

David W. Wilson, Antti Karttunen, Oct 24 1999

Keywords

• nonn
• base
• easy

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].

Examples

			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
		

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.

Crossrefs

Complement: A249719.

Terms of A249721 halved.

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

Programs

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

PARI

is(n)=my(v=digits(n,3)); for(i=1,#v, if(v[i]==2, if(i>1 && v[i-1], return(0)); for(j=i+1,#v, if(v[j]==2, return(0))); return(1))); 1 \\ Charles R Greathouse IV, Feb 23 2024

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

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