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A007931
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Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.
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19
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1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 11111, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Logic-binary sequence: prefix it by the empty word to have all binary words on the alphabet {1,2}.
The least binary word of length k is a(2^k - 1).
See Mathematica program for logic-binary sequence using (0,1) in place of (1,2); the sequence starts with 0,1,00,01,10. - Clark Kimberling, Feb 9 2012
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REFERENCES
| K. Atanassov, On the 97-th, 98-th and the 99-th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
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LINKS
| K. Atanassov, On Some of Smarandache's Problems
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995.
R. R. Forslund, Positive Integer Pages
F. Smarandache, Only Problems, Not Solutions!.
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
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FORMULA
| To get a(n), write n+1 in base 2, remove initial 1, add 1 to all remaining digits: e.g. 11 in base 2 is 1011; remove initial 1 and add 1 to remaining digits: a(10)=122. - Clark Kimberling (ck6(AT)evansville.edu), Mar 11 2003
a(n) = A053645(n+1)+A002275(A000523(n)) = a(n-2^b(n))+10^b(n) where b(n) = A059939(n) = floor[log2(n+1)-1] - Henry Bottomley (se16(AT)btinternet.com), Feb 14 2001
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MATHEMATICA
| f[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[f, 42] (* Robert G. Wilson v Sep 14 2006 *)
(* Next, A007931 using (0, 1) instead of (1, 2) *)
d[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1];
Array[FromCharacterCode[ToCharacterCode[ToString[d[#]]] - 1] &, 100]
(* Peter Moses, at request of Clark Kimberling, Feb 9 2012 *)
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CROSSREFS
| Cf. A007931, A059893, A007932, A045670, A052382, A059939, A059941, A059943, A032924.
Sequence in context: A038115 A089604 A038114 * A136407 A136999 A053880
Adjacent sequences: A007928 A007929 A007930 * A007932 A007933 A007934
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KEYWORD
| nonn,base,nice,easy,changed
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AUTHOR
| R. Muller
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