cp's OEIS Frontend

A283962 Interspersion of the signature sequence of sqrt(1/2).

%I A283962 #21 May 10 2025 10:55:19
%S A283962 1,2,3,4,5,6,7,8,9,11,10,12,13,15,17,14,16,18,20,22,25,19,21,23,26,28,
%T A283962 31,34,24,27,29,32,35,38,41,44,30,33,36,39,42,46,49,52,56,37,40,43,47,
%U A283962 50,54,58,61,65,69,45,48,51,55,59,63,67,71,75,79,84,53
%N A283962 Interspersion of the signature sequence of sqrt(1/2).
%C A283962 Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of (column 1 = A022776).
%C A283962 R(n,m) = position of n*r + m when all the numbers k*r + h, where r = sqrt(2), k >= 1, h >= 0, are jointly ranked. - _Clark Kimberling_, Oct 06 2017
%H A283962 Clark Kimberling, <a href="/A283962/b283962.txt">Antidiagonals n = 1..60, flattened </a>
%H A283962 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%F A283962 R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
%e A283962 Northwest corner of R:
%e A283962    1   2   4   7  10  14  19  24  30
%e A283962    3   5   8  12  16  21  27  33  40
%e A283962    6   9  13  18  23  29  36  43  51
%e A283962   11  15  20  26  32  39  47  44  64
%e A283962   17  22  28  35  42  50  59  68  78
%e A283962   25  31  38  46  54  63  73  83  94
%t A283962 r = Sqrt[1/2]; z = 100;
%t A283962 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283962 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022775, col 1 of A283962 *)
%t A283962 v = Table[s[n], {n, 0, z}] (* A022776, row 1 of A283962*)
%t A283962 w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t A283962 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283962, array *)
%t A283962 Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283962, sequence *)
%o A283962 (PARI)
%o A283962 r = sqrt(1/2);
%o A283962 z = 100;
%o A283962 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283962 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283962 u = v = vector(z + 1);
%o A283962 for(n=1, 101, (v[n] = s(n - 1)));
%o A283962 for(n=1, 101, (u[n] = p(n - 1)));
%o A283962 w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o A283962 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
%o A283962 tabl(10) \\ _Indranil Ghosh_, Mar 21 2017
%o A283962 (Python)
%o A283962 r = 0.5 ** 0.5
%o A283962 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*r)
%o A283962 def p(n): return n + 1 + sum([int((n - k)/r) for k in range(0, n+1)])
%o A283962 v=[s(n) for n in range(0, 101)]
%o A283962 u=[p(n) for n in range(0, 101)]
%o A283962 def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A283962 for n in range(1, 11):
%o A283962     print([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 21 2017
%o A283962 (Python)
%o A283962 import numpy as np
%o A283962 r = np.sqrt(1/2)
%o A283962 x = np.arange(11)
%o A283962 u = np.cumsum(np.ceil(x / r)).astype(int)
%o A283962 v = np.cumsum(np.ceil(x * r)).astype(int)
%o A283962 print(*[1 + u[k] + v[n-k] + k*(n-k) for n in range(11) for k in range(n+1)], sep=', ')
%o A283962 # _David Radcliffe_, May 10 2025
%Y A283962 Cf. A010503, A022775, A022776, A283939.
%K A283962 nonn,tabl,easy
%O A283962 1,2
%A A283962 _Clark Kimberling_, Mar 19 2017