OFFSET
0,2
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
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.
FORMULA
R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
EXAMPLE
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).
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Sep 09 2003
EXTENSIONS
Updated by Clark Kimberling, Sep 23 2014
STATUS
approved