A283938 Interspersion of the signature sequence of tau^2, where tau = (1 + sqrt(5))/2 = golden ratio.

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

Author

Clark Kimberling, Mar 18 2017

Keywords

• nonn
• tabl
• easy

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.

Examples

			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)
		

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. A118276, A283961, A283968, A283969.

Programs

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 *)

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