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A185180 Enumeration table T(n,k) by antidiagonals. The order of the list is symmetrical movement from center to edges diagonal.

%I A185180 #35 Feb 16 2025 08:33:13
%S A185180 1,2,3,5,4,6,9,7,8,10,14,12,11,13,15,20,18,16,17,19,21,27,25,23,22,24,
%T A185180 26,28,35,33,31,29,30,32,34,36,44,42,40,38,37,39,41,43,45,54,52,50,48,
%U A185180 46,47,49,51,53,55,65,63,61,59,57,56,58,60,62,64,66,77,75
%N A185180 Enumeration table T(n,k) by antidiagonals. The order of the list is symmetrical movement from center to edges diagonal.
%C A185180 The natural numbers are grouped in chunks of 1, 2, 3, 4,... as (1), (2,3), (4,5,6), (7,8,9,10), etc and each group fills a diagonal in the table. The smallest number in a group is in A000124, the largest in A000217. Numbers in a group are placed on free spots as close as possible to the middle of the diagonal, given preference to the smaller row numbers in the table if there is a draw.
%C A185180 The resulting array is apparently a transposed version of A064789 (if this was also written as an array).
%C A185180 The order of the list table T(n,k):
%C A185180 if n is odd:
%C A185180 T (floor(n+1)/2,floor(n+1)/2), T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1)
%C A185180 if n is even:
%C A185180 T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1).
%C A185180 Permutation of the natural numbers.
%C A185180 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.
%H A185180 Boris Putievskiy, <a href="/A185180/b185180.txt">Rows n = 1..140 of triangle, flattened</a>
%H A185180 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%H A185180 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A185180 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A185180 a(n) = (i*(i+1) + (j-1)*(j+2*i-4))/2, if j<=i, a(n)=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1, if j>i, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2].
%e A185180 The start of the sequence as table:
%e A185180   1....2....5....9...14...20...27 ...
%e A185180   3....4....7...12...18...25...33 ...
%e A185180   6....8...11...16...23...31...40 ...
%e A185180   10..13...17...22...29...38...48 ...
%e A185180   15..19...24...30...37...46...57 ...
%e A185180   21..26...32...39...47...56...67 ...
%e A185180   28..34...41...49...58...68...79 ...
%e A185180   ...
%e A185180 The start of the sequence as triangle array read by rows:
%e A185180   1;
%e A185180   2,   3;
%e A185180   5,   4,  6;
%e A185180   9,   7,  8, 10;
%e A185180   14, 12, 11, 13, 15;
%e A185180   20, 18, 16, 17, 19, 21;
%e A185180   27, 25, 23, 22, 24, 26, 28;
%e A185180 . . .
%e A185180 Row number k (k > 1) of the triangle contains a permutation of the set of k numbers from (k^2-k+2)/2, (k^2-k+2)/2 + 1 ,...up to (k^2+k-2)/2 + 1, namely (k^2+k-2)/2, (k^2+k-2)/2 -2,...,(k^2-k+2)/2, (k^2-k+2)/2 + 2,..., (k^2+k-2)/2-1, (k^2+k-2)/2+1.
%t A185180 a[n_] := Module[{i, j, t}, i = n - t(t+1)/2; j = (t^2 + 3t + 4)/2 - n; t = Floor[(-1 + Sqrt[8n - 7])/2]; If[j <= i, (i(i+1) + (j-1)(j + 2i - 4))/2, (i(i+1) + (j-1)(j + 2i - 4))/2 + 2(j-i) - 1]];
%t A185180 Array[a, 68] (* _Jean-François Alcover_, Nov 21 2018, from Python *)
%o A185180 (Python)
%o A185180 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A185180 i=n-t*(t+1)/2
%o A185180 j=(t*t+3*t+4)/2-n
%o A185180 if j<=i:
%o A185180    m=(i*(i+1) + (j-1)*(j+2*i-4))/2
%o A185180 else:
%o A185180    m=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1
%Y A185180 Cf. A056011, A056023, A057027, A057028, A064578, A194981, A194982, A188568.
%K A185180 nonn,tabl
%O A185180 1,2
%A A185180 _Boris Putievskiy_, Dec 26 2012