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A004738
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Concatenation of sequences (1,2,...,n-1,n,n-1,...,2) for n >= 2.
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12
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1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9
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OFFSET
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1,2
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COMMENTS
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Also concatenation of sequences n,n-1,...,2,1,2,...,n-1,n.
Table T(n,k) n, k > 0, T(n,k) = n-k+1, if n >= k, T(n,k) = k-n+1, if n < k. Table read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). General case A209301. Let m be a natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A004739, for m=2 the result is A004738, for m=3 the result is A209301. - Boris Putievskiy, Jan 24 2013
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REFERENCES
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F. Smarandache, "Numerical Sequences", University of Craiova, 1975; [ See Arizona State University, Special Collection, Tempe, AZ, USA ].
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LINKS
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FORMULA
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a(n) = floor(sqrt(n) + 1/2) + 1 - abs(n - 1 - (floor(sqrt(n) + 1/2))^2). - Benoit Cloitre, Feb 08 2003
For the general case, a(n) = m*v + (2*v-1)*(t*t-n) + t, where t = floor((sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1.
For m=2, a(n) = 2*v + (2*v-1)*(t*t-n)+t, where t = floor((sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1. (End)
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EXAMPLE
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The start of the sequence as table:
1, 2, 3, 4, 5, 6, 7, ...
2, 1, 2, 3, 4, 5, 6, ...
3, 2, 1, 2, 3, 4, 5, ...
4, 3, 2, 1, 2, 3, 4, ...
5, 4, 3, 2, 1, 2, 3, ...
6, 5, 4, 3, 2, 1, 2, ...
7, 6, 5, 4, 3, 2, 1, ...
...
The start of the sequence as triangle array read by rows:
1;
2, 1, 2;
3, 2, 1, 2, 3;
4, 3, 2, 1, 2, 3, 4;
5, 4, 3, 2, 1, 2, 3, 4, 5;
6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6;
7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7;
...
Row number r contains 2*r - 1 numbers: r, r-1, ..., 1, 2, ..., r. (End)
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MAPLE
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local tri ;
tri := floor(sqrt(n)+1/2) ;
tri+1-abs(n-1-tri^2) ;
end proc:
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MATHEMATICA
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row[n_] := Range[n, 1, -1] ~Join~ Range[2, n];
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PROG
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(PARI) a(n)= floor(sqrt(n)+1/2)+1-abs(n-1-(floor(sqrt(n)+1/2)-1/2)^2)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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R. Muller
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EXTENSIONS
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STATUS
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approved
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