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A165300
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a(n) is the smallest number not already present that permits the cyclic repetition of the path 1,2 of the digits in the sequence.
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7
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1, 2, 12, 121, 21, 212, 1212, 12121, 2121, 21212, 121212, 1212121, 212121, 2121212, 12121212, 121212121, 21212121, 212121212, 1212121212, 12121212121, 2121212121, 21212121212, 121212121212, 1212121212121, 212121212121
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OFFSET
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1,2
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COMMENTS
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Conjecture. (1) If n > 1, and a(n) begins and ends with 1, then a(n+1) is obtained by deleting the initial 1 of a(n); (2) if a(n) begins with 1 and ends with 2 then a(n+1) is obtained by adding a final 1 to a(n); (3) if a(n) begins with 2 and ends with 1 then a(n+1) is obtained by adding a final 2 to a(n); (4) if a(n) begins and ends with 2 then a(n+1) is obtained by adding an initial 1 to a(n). This has been confirmed through a(140), which has 71 digits (and should be fairly easy to prove). - John W. Layman, Sep 22 2009
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LINKS
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FORMULA
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a(n+1) = (1/24)*((a(n) + 10^floor(1 + log_10(a(n))))*(((n-2) mod 4) + ((n-1) mod 4) + 7*(n mod 4) - 5*((n+1) mod 4)) + (10*a(n)+1)*(((n-2) mod 4) + 7*((n-1) mod 4) - 5*(n mod 4) + ((n+1) mod 4)) + (a(n) - 10^floor(log_10(a(n))))*(7*((n-2) mod 4) - 5*((n-1) mod 4) + (n mod 4) + ((n+1) mod 4)) + (10*a(n) + 2)*(-5*((n-2) mod 4) + ((n-1) mod 4) + (n mod 4) + 7*((n+1) mod 4))), with n >= 3 and a(1)=1, a(2)=2. - Paolo P. Lava, Oct 02 2009
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EXAMPLE
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Starting from 1,2 the next number must be 12 because after 1,2 we shall continue with a 1. But 1 is already in the sequence so we need to add a 2 -> 12. And so on.
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MAPLE
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P:=proc(i) local a, n; a:=2; print(1); print(2); for n from 3 by 1 to i do a:=1/24*((a+10^floor(1+evalf(log10(a), 100)))*(((n-2) mod 4)+((n-1) mod 4)+7*(n mod 4)-5*((n+1) mod 4))+(10*a+1)*(((n-2) mod 4)+7*((n-1) mod 4)-5*(n mod 4)+((n+1) mod 4))+(a-10^floor(evalf(log10(a), 100)))*(7*((n-2) mod 4)-5*((n-1) mod 4)+(n mod 4)+((n+1) mod 4))+(10*a+2)*(-5*((n-2) mod 4)+((n-1) mod 4)+(n mod 4)+7*((n+1) mod 4))); print(a); od; end: P(200); # Paolo P. Lava, Oct 02 2009
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CROSSREFS
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KEYWORD
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easy,base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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