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r = 2040; r1 = 12; (*r=# rows of T, r1=# rows to show*);

c = 2040; c1 = 12; (*c=# cols of T, c1=# cols to show*);

u = Table[s[n], {n, 0, 100400}] (* A283733 *)

TableForm[Table[w[i, j], {i, 1, 10r1}, {j, 1, 10c1}]] (* A283734, array *)

TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10r1}, {j, 1, 10c1}]] (* A283734, array, by formula *)

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Rank array, R, of the golden ratio, read by antidiagonals downwards.

Northwest The corner of R begins:

allocated for Clark KimberlingRank array, R, of the golden ratio, read by antidiagonals.

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 9, 11, 14, 16, 19, 13, 15, 18, 21, 24, 28, 17, 20, 23, 26, 30, 34, 38, 22, 25, 29, 32, 36, 41, 45, 50, 27, 31, 35, 39, 43, 48, 53, 58, 63, 33, 37, 42, 46, 51, 56, 61, 67, 72, 78, 40, 44, 49, 54, 59, 65, 70, 76, 82, 88, 95, 47

1,2

Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of column 1; column 1 is given by r(n) = r(n-1) + 1 + L(n), where L = lower Wythoff sequence (A000201).

Clark Kimberling, <a href="/A283734/b283734.txt">Antidiagonals n = 1..60, flattened</a>

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.

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

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

28 34 41 48 56 65 74 84

38 45 53 61 70 80 90 101

50 58 67 76 86 97 108 120

Let t = golden ratio = (1 + sqrt(5))/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).

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[n*GoldenRatio];

u = Table[s[n], {n, 0, 100}] (* A283733 *)

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}]] (* A283734, array *)

Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A283734, sequence *)

TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A283734, array, by formula *)

Cf. A001622, A255977 (row 1), A283733 (column 1), A000201, A087465.

allocated

nonn,tabl,easy

Clark Kimberling, Mar 16 2017

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allocated for Clark Kimberling

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