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 A081344 Natural numbers in square maze arrangement, read by antidiagonals. 15
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 For generalizations see A219159, A213928. - Boris Putievskiy, Mar 10 2013 LINKS Boris Putievskiy, Rows n = 1..100 of triangle, flattened Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. Eric Weisstein's World of Mathematics, Pairing functions FORMULA From Boris Putievskiy, Dec 19 2012: (Start) 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) EXAMPLE 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 MATHEMATICA 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 *) 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 j >= i:      m=(j-1)**2 + j + (j-i)*(-1)**(j-1) else:      m=(i-1)**2 + i - (i-j)*(-1)**(i-1) # Boris Putievskiy, Dec 19 2012 CROSSREFS Cf. A219159, A213928. The main diagonal is A002061. The following appear within interlaced sequences: A016754, A001844, A053755, A004120. The first row is A081345. The first column is A081346. The inverse permutation A194280, the first inverse function (numbers of rows) A220603, the second inverse function (numbers of columns) A220604. Sequence in context: A133057 A155523 A019912 * A227272 A021405 A201946 Adjacent sequences:  A081341 A081342 A081343 * A081345 A081346 A081347 KEYWORD nonn,tabl AUTHOR Paul Barry, Mar 19 2003 STATUS approved

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Last modified October 23 12:19 EDT 2019. Contains 328345 sequences. (Running on oeis4.)