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A213922
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.
3
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
OFFSET
1,2
COMMENTS
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);
...
FORMULA
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).
EXAMPLE
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;
...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.
Sequence in context: A346411 A199618 A088745 * A306568 A154743 A020812
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Mar 05 2013
STATUS
approved