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).

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, 33, 37, 22, 25, 29, 32, 36, 40, 44, 48, 28, 31, 35, 39, 43, 47, 52, 56, 61, 34, 38, 42, 46, 51, 55, 60, 65, 70, 75, 41, 45, 50, 54, 59, 64, 69, 74, 80, 85, 91, 49, 53, 58, 63, 68, 73

Offset: 0

Author

Clark Kimberling, Sep 09 2003

Keywords

• nonn
• tabl

Comments

The sequence is a permutation of the positive integers and the array is a transposable dispersion.

Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here:

1 2 4 8 16 32

3 6 12 24 48 96

9 18 36 72 144 288

27 54 108 216 432 864

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

Examples

			Northwest corner of R:
   1    2    4    6    9   13   17   22
   3    5    8   11   15   20   25   31
   7   10   14   18   23   29   35   42
  12   16   21   26   32   39   46   54
  19   24   30   36   43   51   59   68
  27   33   40   47   55   64   73   83
  37   44   52   60   69   79   89  100
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).
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened
• Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Crossrefs

Cf. A087466, A087468, A087483, A007780 (row 1), A077043 (column 1).

Programs

Mathematica

r = 20; r1 = 12;(*r=# rows of T,r1=# rows to show*);
c = 20; c1 = 12;(*c=# cols of T,c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
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]];
TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]]   (* A087465 array *)
Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *)
TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)

Formula

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.

Extensions

Updated by Clark Kimberling, Sep 23 2014