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A283939 Interspersion of the signature sequence of sqrt(2).

%I A283939 #14 May 10 2025 09:15:44
%S A283939 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 A283939 21,19,44,41,38,35,32,29,27,24,56,52,49,46,42,39,36,33,30,69,65,61,58,
%U A283939 54,50,47,43,40,37,84,79,75,71,67,63,59,55,51,48,45,100
%N A283939 Interspersion of the signature sequence of sqrt(2).
%C A283939 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 A283939 Clark Kimberling, <a href="/A283939/b283939.txt">Antidiagonals n = 1..60, flattened</a>
%H A283939 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 A283939 Northwest corner:
%e A283939   1   3   6   11   17   25   34   44   56
%e A283939   2   5   9   15   22   31   41   52   65
%e A283939   4   8   13  20   28   38   49   61   75
%e A283939   7   12  18  26   35   46   58   71   86
%e A283939   10  16  23  32   42   54   67   81   97
%e A283939   14  21  29  39   50   63   77   91   109
%t A283939 r = Sqrt[2]; z = 100;
%t A283939 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283939 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022776, col 1 of A283939 *)
%t A283939 v = Table[s[n], {n, 0, z}] (* A022775, row 1 of A283939*)
%t A283939 w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t A283939 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)
%t A283939 p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)
%o A283939 (PARI)
%o A283939 r = sqrt(2);
%o A283939 z = 100;
%o A283939 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283939 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283939 u = v = vector(z + 1);
%o A283939 for(n=1, 101, (v[n] = s(n - 1)));
%o A283939 for(n=1, 101, (u[n] = p(n - 1)));
%o A283939 w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o A283939 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
%o A283939 tabl(10) \\ _Indranil Ghosh_, Mar 21 2017
%o A283939 (Python)
%o A283939 sqrt2 = 2 ** 0.5
%o A283939 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*sqrt2)
%o A283939 def p(n): return n + 1 + sum([int((n - k)/sqrt2) for k in range(0, n+1)])
%o A283939 v=[s(n) for n in range(0, 101)]
%o A283939 u=[p(n) for n in range(0, 101)]
%o A283939 def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A283939 for n in range(1, 11):
%o A283939     print ([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 21 2017
%Y A283939 Cf. A007336, A002193, A022776, A283962.
%K A283939 nonn,tabl,easy
%O A283939 1,2
%A A283939 _Clark Kimberling_, Mar 19 2017