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%I #26 Jan 26 2018 05:20:26
%S 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 33,37,22,25,29,32,36,40,44,48,28,31,35,39,43,47,52,56,61,34,38,42,46,
%U 51,55,60,65,70,75,41,45,50,54,59,64,69,74,80,85,91,49,53,58,63,68,73
%N 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 The sequence is a permutation of the positive integers and the array is a transposable dispersion.
%C Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here:
%C 1 2 4 8 16 32
%C 3 6 12 24 48 96
%C 9 18 36 72 144 288
%C 27 54 108 216 432 864
%C 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 Clark Kimberling, <a href="/A087465/b087465.txt">Antidiagonals n = 1..60, flattened</a>
%H 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 R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
%e Northwest corner of R:
%e 1 2 4 6 9 13 17 22
%e 3 5 8 11 15 20 25 31
%e 7 10 14 18 23 29 35 42
%e 12 16 21 26 32 39 46 54
%e 19 24 30 36 43 51 59 68
%e 27 33 40 47 55 64 73 83
%e 37 44 52 60 69 79 89 100
%e 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 r = 20; r1 = 12;(*r=# rows of T,r1=# rows to show*);
%t c = 20; c1 = 12;(*c=# cols of T,c1=# cols to show*);
%t s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
%t v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
%t 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 TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A087465 array *)
%t Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *)
%t TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)
%Y Cf. A087466, A087468, A087483, A007780 (row 1), A077043 (column 1).
%K nonn,tabl
%O 0,2
%A _Clark Kimberling_, Sep 09 2003
%E Updated by _Clark Kimberling_, Sep 23 2014
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