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A081344
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Natural numbers in square maze arrangement, read by antidiagonals.
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17
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1, 2, 4, 9, 3, 5, 10, 8, 6, 16, 25, 11, 7, 15, 17, 26, 24, 12, 14, 18, 36, 49, 27, 23, 13, 19, 35, 37, 50, 48, 28, 22, 20, 34, 38, 64, 81, 51, 47, 29, 21, 33, 39, 63, 65, 82, 80, 52, 46, 30, 32, 40, 62, 66, 100, 121, 83, 79, 53, 45, 31, 41, 61, 67, 99, 101, 122, 120, 84, 78, 54
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OFFSET
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1,2
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COMMENTS
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Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).
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. - Boris Putievskiy, Dec 16 2012
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LINKS
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FORMULA
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a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,
a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-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). (End)
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EXAMPLE
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The start of the sequence as table T(i,j), i,j > 0:
1 .. 4 .. 5 .. 16...
2 .. 3 .. 6 .. 15...
9 .. 8 .. 7 .. 14...
10..11 ..12 .. 13...
. . .
Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,i)=(i-1)^2+i + (i-j)*(-1)^(i-1), if i < j: T(i,j)=(j-1)^2+j - (j-i)*(-1)^(j-1). - Boris Putievskiy, Dec 19 2012
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MATHEMATICA
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T[n_, k_] := T[n, k] = Which[OddQ[n] && k==1, n^2, EvenQ[k] && n==1, k^2, EvenQ[n] && k==1, T[n-1, 1]+1, OddQ[k] && n==1, T[1, k-1]+1, k <= n, T[n, k-1]+1 - 2 Mod[n, 2], True, T[n-1, k]-1 + 2 Mod[k, 2]]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 20 2019 *)
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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 j >= i:
m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
else:
m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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