OFFSET
1,2
COMMENTS
Row n is the ordered sequence of numbers k such that A023133(k) = n. As a sequence, it 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.
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.
EXAMPLE
Northwest corner:
1 5 12 22 35 51 70 92 118
2 7 15 26 40 57 77 100 127
3 9 18 30 45 63 84 108 136
4 11 21 34 50 69 91 115 145
6 14 25 39 56 76 99 125 155
8 17 29 44 62 83 107 134 165
MATHEMATICA
PROG
(PARI)
\\ Produces the triangle when the array is read by antidiagonals
r = Pi;
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 26 2017
(Python)
# Produces the triangle when the array is read by antidiagonals
import math
from mpmath import *
mp.dps = 100
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*pi))
def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) 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 26 2017
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Mar 26 2017
STATUS
approved