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 A209293 Inverse permutation of A185180. 5
 1, 2, 3, 5, 4, 6, 8, 9, 7, 10, 13, 12, 14, 11, 15, 18, 19, 17, 20, 16, 21, 25, 24, 26, 23, 27, 22, 28, 32, 33, 31, 34, 30, 35, 29, 36, 41, 40, 42, 39, 43, 38, 44, 37, 45, 50, 51, 49, 52, 48, 53, 47, 54, 46, 55, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67 (list; table; graph; refs; listen; history; text; internal format)
 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. Enumeration table T(n,k) by diagonals. The order of the list if n is odd  - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1). if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1). Table T(n,k) contains: Column number 1 A000217, column number 2 A000124, column number 3 A000096, column number 4 A152948, column number 5 A034856, column number 6 A152950, column number 7 A055998. Row    numder 1 A000982, row    number 2 A097063. LINKS Boris Putievskiy, Rows n = 1..140 of triangle, flattened Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO] Eric W. Weisstein, MathWorld: Pairing functions FORMULA As table T(n,k) read by antidiagonals T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0. As linear sequence a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where m1 = int((i+j)/2)+int(i/2)*(-1)^(i+t+1), m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t), t = int((math.sqrt(8*n-7) - 1)/ 2), i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n. EXAMPLE The start of the sequence as table: 1....2...5...8..13..18...25...32...41... 3....4...9..12..19..24...33...40...51... 6....7..14..17..26..31...42...49...62... 10..11..20..23..34..39...52...59...74... 15..16..27..30..43..48...63...70...87... 21..22..35..38..53..58...75...82..101... 28..29..44..47..64..69...88...95..116... 36..37..54..57..76..81..102..109..132... 45..46..65..68..89..94..117..124..149... . . . The start of the sequence as triangle array read by rows: 1; 2,3; 5,4,6; 8,9,7,10; 13,12,14,11,15; 18,19,17,20,16,21; 25,24,26,23,27,22,28; 32,33,31,34,30,35,29,36; 41,40,42,39,43,38,44,37,45; . . . Row number r contains permutation from r numbers: if r is odd  ceiling(r^2/2), ceiling(r^2/2)+1, ceiling(r^2/2)-1, ceiling(r^2/2)+2, ceiling(r^2/2)-2,...r*(r+1)/2; if r is even ceiling(r^2/2), ceiling(r^2/2)-1, ceiling(r^2/2)+1, ceiling(r^2/2)-2, ceiling(r^2/2)+2,...r*(r+1)/2; MATHEMATICA max = 10; row[n_] := Table[Ceiling[(n + k - 1)^2/2] + If[OddQ[k], 1, -1]*Floor[n/2], {k, 1, max}]; t = Table[row[n], {n, 1, max}]; Table[t[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 17 2013 *) 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 m1=int((i+j)/2)+int(i/2)*(-1)**(i+t+1) m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t) m=(m1+m2-1)*(m1+m2-2)/2+m1 CROSSREFS Cf. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063. Sequence in context: A073291 A073280 A073884 * A328021 A097290 A279344 Adjacent sequences:  A209290 A209291 A209292 * A209294 A209295 A209296 KEYWORD nonn,tabl AUTHOR Boris Putievskiy, Jan 16 2013 STATUS approved

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Last modified January 22 07:30 EST 2020. Contains 331139 sequences. (Running on oeis4.)