OFFSET
1,2
COMMENTS
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 3 7 13 20 29 40 53
2 5 10 17 25 35 47 61
4 8 14 22 31 42 55 70
6 11 18 27 37 49 63 79
9 15 23 33 44 57 72 89
12 19 28 39 51 65 81 99
16 24 34 46 59 74 91 110
21 30 41 54 68 84 102 122
MATHEMATICA
PROG
(PARI)
r = sqrt(3);
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(3)))
def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(3))) 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