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A283940 Interspersion of the signature sequence of sqrt(3).

%I A283940 #15 May 10 2025 11:28:40
%S A283940 1,3,2,7,5,4,13,10,8,6,20,17,14,11,9,29,25,22,18,15,12,40,35,31,27,23,
%T A283940 19,16,53,47,42,37,33,28,24,21,67,61,55,49,44,39,34,30,26,83,76,70,63,
%U A283940 57,51,46,41,36,32,101,93,86,79,72,65,59,54,48,43,38
%N A283940 Interspersion of the signature sequence of sqrt(3).
%C A283940 Row n is the ordered sequence of numbers k such that A007337(k)=n.  As a sequence, A283940 is a permutation of the positive integers.  This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.
%H A283940 Clark Kimberling, <a href="/A283940/b283940.txt">Antidiagonals n = 1..60, flattened</a>
%H A283940 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 A283940 Northwest corner:
%e A283940   1   3    7    13   20   29   40   53
%e A283940   2   5    10   17   25   35   47   61
%e A283940   4   8    14   22   31   42   55   70
%e A283940   6   11   18   27   37   49   63   79
%e A283940   9   15   23   33   44   57   72   89
%e A283940   12  19   28   39   51   65   81   99
%e A283940   16  24   34   46   59   74   91   110
%e A283940   21  30   41   54   68   84   102  122
%t A283940 r = Sqrt[3]; z = 100;
%t A283940 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283940 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022778, col 1 of A283940 *)
%t A283940 v = Table[s[n], {n, 0, z}] (* A022777, row 1 of A283940*)
%t A283940 w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t A283940 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283940, array *)
%t A283940 Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283940, sequence *)
%o A283940 (PARI)
%o A283940 r = sqrt(3);
%o A283940 z = 100;
%o A283940 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283940 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283940 u = v = vector(z + 1);
%o A283940 for(n=1, 101, (v[n] = s(n - 1)));
%o A283940 for(n=1, 101, (u[n] = p(n - 1)));
%o A283940 w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o A283940 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
%o A283940 tabl(10) \\ _Indranil Ghosh_, Mar 21 2017
%o A283940 (Python)
%o A283940 r = 3 ** 0.5
%o A283940 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*r)
%o A283940 def p(n): return n + 1 + sum([int((n - k)/r) for k in range(0, n+1)])
%o A283940 v=[s(n) for n in range(0, 101)]
%o A283940 u=[p(n) for n in range(0, 101)]
%o A283940 def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A283940 for n in range(1, 11):
%o A283940     print ([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 21 2017
%o A283940 (Python)
%o A283940 import numpy as np
%o A283940 r = np.sqrt(3)
%o A283940 x = np.arange(11)
%o A283940 u = np.cumsum(np.ceil(x / r)).astype(int)
%o A283940 v = np.cumsum(np.ceil(x * r)).astype(int)
%o A283940 print(*[1 + u[k] + v[n-k] + k*(n-k) for n in range(11) for k in range(n+1)], sep=', ')
%o A283940 # _David Radcliffe_, May 10 2025
%Y A283940 Cf. A002194, A007337, A022777, A022778.
%K A283940 nonn,tabl,easy
%O A283940 1,2
%A A283940 _Clark Kimberling_, Mar 19 2017