# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a209293 Showing 1-1 of 1 %I A209293 #21 Nov 29 2023 06:57:49 %S A209293 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, %T A209293 22,28,32,33,31,34,30,35,29,36,41,40,42,39,43,38,44,37,45,50,51,49,52, %U A209293 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 %N A209293 Inverse permutation of A185180. %C A209293 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. %C A209293 Enumeration table T(n,k) by diagonals. The order of the list %C A209293 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). %C A209293 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). %C A209293 Table T(n,k) contains: %C A209293 Column number 1 A000217, %C A209293 column number 2 A000124, %C A209293 column number 3 A000096, %C A209293 column number 4 A152948, %C A209293 column number 5 A034856, %C A209293 column number 6 A152950, %C A209293 column number 7 A055998. %C A209293 Row numder 1 A000982, %C A209293 row number 2 A097063. %H A209293 Boris Putievskiy, Rows n = 1..140 of triangle, flattened %H A209293 Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. %H A209293 Eric Weisstein's World of Mathematics, Pairing functions %H A209293 Index entries for sequences that are permutations of the natural numbers %F A209293 As table T(n,k) read by antidiagonals %F A209293 T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0. %F A209293 As linear sequence %F A209293 a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where %F A209293 m1 = int((i+j)/2)+int(i/2)*(-1)^(i+t+1), %F A209293 m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t), %F A209293 t = int((math.sqrt(8*n-7) - 1)/ 2), %F A209293 i = n-t*(t+1)/2, %F A209293 j = (t*t+3*t+4)/2-n. %e A209293 The start of the sequence as table: %e A209293 1....2...5...8..13..18...25...32...41... %e A209293 3....4...9..12..19..24...33...40...51... %e A209293 6....7..14..17..26..31...42...49...62... %e A209293 10..11..20..23..34..39...52...59...74... %e A209293 15..16..27..30..43..48...63...70...87... %e A209293 21..22..35..38..53..58...75...82..101... %e A209293 28..29..44..47..64..69...88...95..116... %e A209293 36..37..54..57..76..81..102..109..132... %e A209293 45..46..65..68..89..94..117..124..149... %e A209293 . . . %e A209293 The start of the sequence as triangle array read by rows: %e A209293 1; %e A209293 2,3; %e A209293 5,4,6; %e A209293 8,9,7,10; %e A209293 13,12,14,11,15; %e A209293 18,19,17,20,16,21; %e A209293 25,24,26,23,27,22,28; %e A209293 32,33,31,34,30,35,29,36; %e A209293 41,40,42,39,43,38,44,37,45; %e A209293 . . . %e A209293 Row number r contains permutation from r numbers: %e A209293 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; %e A209293 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; %t A209293 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 *) %o A209293 (Python) %o A209293 t=int((math.sqrt(8*n-7) - 1)/ 2) %o A209293 i=n-t*(t+1)/2 %o A209293 j=(t*t+3*t+4)/2-n %o A209293 m1=int((i+j)/2)+int(i/2)*(-1)**(i+t+1) %o A209293 m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t) %o A209293 m=(m1+m2-1)*(m1+m2-2)/2+m1 %Y A209293 Cf. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063. %K A209293 nonn,tabl %O A209293 1,2 %A A209293 _Boris Putievskiy_, Jan 16 2013 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE