%I #8 Dec 07 2019 12:18:28
%S 1,3,2,6,5,4,11,9,8,7,17,15,13,12,10,25,22,20,18,16,14,34,31,28,26,23,
%T 21,19,44,41,38,35,32,29,27,24,56,52,49,46,42,39,36,33,30,69,65,61,58,
%U 54,50,47,43,40,37,84,79,75,71,67,63,59,55,51,48,45,100
%N Interspersion of the signature sequence of sqrt(2).
%C Row n is the ordered sequence of numbers k such that A007336(k)=n. As a sequence, A283939 is a permutation of the positive integers. As an array, A283939 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = sqrt(2). This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.
%H Clark Kimberling, <a href="/A283939/b283939.txt">Antidiagonals n = 1..60, flattened</a>
%H 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 Northwest corner:
%e 1 3 6 11 17 25 34 44 56
%e 2 5 9 15 22 31 41 52 65
%e 4 8 13 20 28 38 49 61 75
%e 7 12 18 26 35 46 58 71 86
%e 10 16 23 32 42 54 67 81 97
%e 14 21 29 39 50 63 77 91 109
%t r = Sqrt[2]; z = 100;
%t s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022776, col 1 of A283939 *)
%t v = Table[s[n], {n, 0, z}] (* A022775, row 1 of A283939*)
%t w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)
%t p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)
%o (PARI)
%o r = sqrt(2);
%o z = 100;
%o s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o u = v = vector(z + 1);
%o for(n=1, 101, (v[n] = s(n - 1)));
%o for(n=1, 101, (u[n] = p(n - 1)));
%o w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
%o tabl(10) \\ _Indranil Ghosh_, Mar 21 2017
%o (Python)
%o from sympy import sqrt
%o import math
%o def s(n): return 1 if n<1 else s(n - 1) + 1 +
%o int(math.floor(n*sqrt(2)))
%o def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(2))) for k in range(0, n+1)])
%o v=[s(n) for n in range(0, 101)]
%o u=[p(n) for n in range(0, 101)]
%o def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o for n in range(1, 11):
%o ....print [w(k, n - k + 1) for k in range(1, n + 1)] # _Indranil Ghosh_, Mar 21 2017
%Y Cf. A007336, A002193, A022776, A283962.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Mar 19 2017