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A283734
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Rank array, R, of the golden ratio, read by antidiagonals downwards.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
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EXAMPLE
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The corner of R begins:
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).
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MATHEMATICA
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r = 40; r1 = 12; (*r=# rows of T, r1=# rows to show*);
c = 40; 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, 400}] (* 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, r1}, {j, 1, c1}]] (* 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, r1}, {j, 1, c1}]] (* A283734, array, by formula *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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