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

%I A283938 #24 Mar 19 2017 13:18:00
%S A283938 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,
%T A283938 17,12,78,66,55,45,36,28,21,15,99,86,73,62,51,41,33,25,19,123,108,94,
%U A283938 81,69,57,47,38,30,23,150,133,117,103,89,76,64,53,44,35
%N A283938 Interspersion of the signature sequence of tau^2, where tau = (1 + sqrt(5))/2 = golden ratio.
%C A283938 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.
%H A283938 Clark Kimberling, <a href="/A283938/b283938.txt">Antidiagonals n = 1..60, flattened</a>
%H A283938 Clark Kimberling and John E. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%e A283938 Northwest corner:
%e A283938 1   4  10   18  29  43  59   78  99   123
%e A283938 2   6  13   22  34  49  66   86  108  133
%e A283938 3   8  16   26  39  55  73   94  117  143
%e A283938 5  11  20   31  45  62  81  103  127  154
%e A283938 7  14  24   36  51  69  89  112  137  165
%e A283938 9  17  28   41  57  76  97  121  147  176
%e A283938 From _Indranil Ghosh_, Mar 19 2017: (Start)
%e A283938 Triangle formed when the array is read by antidiagonals:
%e A283938     1;
%e A283938     4,   2;
%e A283938    10,   6,  3;
%e A283938    18,  13,  8,  5;
%e A283938    29,  22, 16, 11,  7;
%e A283938    43,  34, 26, 20, 14,  9;
%e A283938    59,  49, 39, 31, 24, 17, 12;
%e A283938    78,  66, 55, 45, 36, 28, 21, 15;
%e A283938    99,  86, 73, 62, 51, 41, 33, 25, 19;
%e A283938   123, 108, 94, 81, 69, 57, 47, 38, 30, 23;
%e A283938   ...
%e A283938 (End)
%t A283938 r = GoldenRatio^2; z = 100;
%t A283938 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283938 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968, row 1 of A283938 *)
%t A283938 v = Table[s[n], {n, 0, z}] (* A283969, col 1 of A283938 *)
%t A283938 w[i_, j_] := v[[i]] + u[[j]] + (i - 1)*(j - 1) - 1;
%t A283938 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283938, array *)
%t A283938 Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283938, sequence *)
%o A283938 (PARI)
%o A283938 \\ This code produces the triangle mentioned in the example section
%o A283938 r = (3 +sqrt(5))/2;
%o A283938 z = 100;
%o A283938 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283938 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283938 u = v = vector(z + 1);
%o A283938 for(n=1, 101, (v[n] = s(n - 1)));
%o A283938 for(n=1, 101, (u[n] = p(n - 1)));
%o A283938 w(i,j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
%o A283938 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k),", ");); print(););};
%o A283938 tabl(10) \\ _Indranil Ghosh_, Mar 19 2017
%Y A283938 Cf. A118276, A283961, A283968, A283969.
%K A283938 nonn,tabl,easy
%O A283938 1,2
%A A283938 _Clark Kimberling_, Mar 18 2017