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A213922
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Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.
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3
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1, 3, 4, 8, 2, 9, 15, 6, 7, 16, 24, 13, 5, 14, 25, 35, 22, 11, 12, 23, 36, 48, 33, 20, 10, 21, 34, 49, 63, 46, 31, 18, 19, 32, 47, 64, 80, 61, 44, 29, 17, 30, 45, 62, 81, 99, 78, 59, 42, 27, 28, 43, 60, 79, 100, 120, 97, 76, 57, 40, 26, 41, 58, 77, 98, 121
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OFFSET
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1,2
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COMMENTS
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Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
...
T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
...
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LINKS
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FORMULA
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As a table,
T(n,k) = n*n - 2*k + 2, if n >= k;
T(n,k) = k*k - 2*n + 1, if n < k.
As a linear sequence,
a(n) = i*i - 2*j + 2, if i >= j;
a(n) = j*j - 2*i + 1, if i < j
where
i = n - t*(t+1)/2,
j = (t*t + 3*t + 4)/2 - n,
t = floor((-1 + sqrt(8*n-7))/2).
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EXAMPLE
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The start of the sequence as a table:
1, 3, 8, 15, 24, 35, ...
4, 2, 6, 13, 22, 33, ...
9, 7, 5, 11, 20, 31, ...
16, 14, 12, 10, 18, 29, ...
25, 23, 21, 19, 17, 27, ...
36, 34, 32, 30, 28, 26, ...
...
The start of the sequence as triangular array read by rows:
1;
3, 4;
8, 2, 9;
15, 6, 7, 16;
24, 13, 5, 14, 25;
35, 22, 11, 12, 23, 36;
...
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MATHEMATICA
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f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Aug 19 2017 *)
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PROG
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(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i >= j:
result=i*i-2*j+2
else:
result=j*j-2*i+1
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CROSSREFS
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Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.
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KEYWORD
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AUTHOR
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STATUS
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approved
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