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A283938
Interspersion of the signature sequence of tau^2, where tau = (1 + sqrt(5))/2 = golden ratio.
3
1, 4, 2, 10, 6, 3, 18, 13, 8, 5, 29, 22, 16, 11, 7, 43, 34, 26, 20, 14, 9, 59, 49, 39, 31, 24, 17, 12, 78, 66, 55, 45, 36, 28, 21, 15, 99, 86, 73, 62, 51, 41, 33, 25, 19, 123, 108, 94, 81, 69, 57, 47, 38, 30, 23, 150, 133, 117, 103, 89, 76, 64, 53, 44, 35
OFFSET
1,2
COMMENTS
Row n is the ordered sequence of numbers k such that A118276(k) = n. As a sequence, A283938 is a permutation of the positive integers. As an array, A283938 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = tau^2 = (3 + sqrt(5))/2. This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.
LINKS
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
EXAMPLE
Northwest corner:
1 4 10 18 29 43 59 78 99 123
2 6 13 22 34 49 66 86 108 133
3 8 16 26 39 55 73 94 117 143
5 11 20 31 45 62 81 103 127 154
7 14 24 36 51 69 89 112 137 165
9 17 28 41 57 76 97 121 147 176
From Indranil Ghosh, Mar 19 2017: (Start)
Triangle formed when the array is read by antidiagonals:
1;
4, 2;
10, 6, 3;
18, 13, 8, 5;
29, 22, 16, 11, 7;
43, 34, 26, 20, 14, 9;
59, 49, 39, 31, 24, 17, 12;
78, 66, 55, 45, 36, 28, 21, 15;
99, 86, 73, 62, 51, 41, 33, 25, 19;
123, 108, 94, 81, 69, 57, 47, 38, 30, 23;
...
(End)
MATHEMATICA
r = GoldenRatio^2; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968, row 1 of A283938 *)
v = Table[s[n], {n, 0, z}] (* A283969, col 1 of A283938 *)
w[i_, j_] := v[[i]] + u[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283938, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283938, sequence *)
PROG
(PARI)
\\ This code produces the triangle mentioned in the example section
r = (3 +sqrt(5))/2;
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k), ", "); ); print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 19 2017
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Mar 18 2017
STATUS
approved