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A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).

%I A087465 #29 Aug 24 2024 22:45:23
%S A087465 1,2,3,4,5,7,6,8,10,12,9,11,14,16,19,13,15,18,21,24,27,17,20,23,26,30,
%T A087465 33,37,22,25,29,32,36,40,44,48,28,31,35,39,43,47,52,56,61,34,38,42,46,
%U A087465 51,55,60,65,70,75,41,45,50,54,59,64,69,74,80,85,91,49,53,58,63,68,73
%N A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).
%C A087465 The sequence is a permutation of the positive integers and the array is a transposable dispersion.
%C A087465 Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here:
%C A087465    1     2     4     8    16    32
%C A087465    3     6    12    24    48    96
%C A087465    9    18    36    72   144   288
%C A087465   27    54   108   216   432   864
%C A087465 Then R(n,k) is the rank of T(n,k) when all the numbers in {T(n,k)} are jointly ranked. - _Clark Kimberling_, Jan 25 2018
%H A087465 Clark Kimberling, <a href="/A087465/b087465.txt">Antidiagonals n = 1..60, flattened</a>
%H A087465 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%F A087465 R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
%e A087465 Northwest corner of R:
%e A087465    1    2    4    6    9   13   17   22
%e A087465    3    5    8   11   15   20   25   31
%e A087465    7   10   14   18   23   29   35   42
%e A087465   12   16   21   26   32   39   46   54
%e A087465   19   24   30   36   43   51   59   68
%e A087465   27   33   40   47   55   64   73   83
%e A087465   37   44   52   60   69   79   89  100
%e A087465 Let t=3/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d.  Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).
%t A087465 r = 20; r1 = 12;(*r=# rows of T,r1=# rows to show*);
%t A087465 c = 20; c1 = 12;(*c=# cols of T,c1=# cols to show*);
%t A087465 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
%t A087465 v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
%t A087465 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,   Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; w[i_, j_] := rows[[i, j]];
%t A087465 TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]]   (* A087465 array *)
%t A087465 Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *)
%t A087465 TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)
%Y A087465 Cf. A087466, A087468, A087483, A007780 (row 1), A077043 (column 1).
%K A087465 nonn,tabl
%O A087465 0,2
%A A087465 _Clark Kimberling_, Sep 09 2003
%E A087465 Updated by _Clark Kimberling_, Sep 23 2014