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 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). 13
 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 (list; graph; refs; listen; history; text; internal format)
 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 (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 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

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Last modified August 14 01:00 EDT 2024. Contains 375146 sequences. (Running on oeis4.)