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

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, 31, 34, 24, 27, 29, 32, 35, 38, 41, 44, 30, 33, 36, 39, 42, 46, 49, 52, 56, 37, 40, 43, 47, 50, 54, 58, 61, 65, 69, 45, 48, 51, 55, 59, 63, 67, 71, 75, 79, 84, 53

Offset: 1

Author

Clark Kimberling, Mar 19 2017

Keywords

• nonn
• tabl
• easy

Comments

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).

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

Examples

			Northwest corner of R:
   1   2   4   7  10  14  19  24  30
   3   5   8  12  16  21  27  33  40
   6   9  13  18  23  29  36  43  51
  11  15  20  26  32  39  47  44  64
  17  22  28  35  42  50  59  68  78
  25  31  38  46  54  63  73  83  94
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened
• Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Crossrefs

Cf. A010503, A022775, A022776, A283939.

Programs

Mathematica

r = Sqrt[1/2]; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022775, col 1 of A283962 *)
v = Table[s[n], {n, 0, z}] (* A022776, row 1 of A283962*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283962, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283962, sequence *)

PARI

r = sqrt(1/2);
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 21 2017

Python

r = 0.5 ** 0.5
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*r)
def p(n): return n + 1 + sum([int((n - k)/r) for k in range(0, n+1)])
v=[s(n) for n in range(0, 101)]
u=[p(n) for n in range(0, 101)]
def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
for n in range(1, 11):
    print([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 21 2017

Python

import numpy as np
r = np.sqrt(1/2)
x = np.arange(11)
u = np.cumsum(np.ceil(x / r)).astype(int)
v = np.cumsum(np.ceil(x * r)).astype(int)
print(*[1 + u[k] + v[n-k] + k*(n-k) for n in range(11) for k in range(n+1)], sep=', ')
# David Radcliffe, May 10 2025

Formula

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.