OFFSET
1,2
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
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.
FORMULA
R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
EXAMPLE
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
MATHEMATICA
PROG
(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)
from sympy import sqrt
import math
def s(n): return 1 if n<1 else s(n - 1) + 1 +
int(math.floor(n*sqrt(1/2)))
def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(1/2))) 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
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Mar 19 2017
STATUS
approved