OFFSET
1,2
COMMENTS
Antidiagonals n = 1..60, flattened.
Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of column 1, where column 1 is given by c(n) = c(n-1) + 1 + U(n), where U = upper Wythoff sequence (A001950).
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1829
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 3 5 7 9 12 15
4 6 8 11 14 17 21 25
10 13 16 20 24 28 33 38
18 22 26 31 36 41 47 53
29 34 39 45 51 57 64 71
43 49 55 62 69 76 84 92
Let t = (golden ratio)^2 = (3 + sqrt(5))/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d. Thus R(2,0) = 10 is the rank of (0,2) in the list (0,0) << (1,0) << (2,0) << (0,1) << (3,0) << (1,1) << (4,0) << (2,1) << (5,0) << (0,2).
From Indranil Ghosh, Mar 19 2017: (Start)
Triangle formed when the array is read by antidiagonals:
1;
2, 4;
3, 6, 10;
5, 8, 13, 18;
7, 11, 16, 22, 29;
9, 14, 20, 26, 34, 43;
12, 17, 24, 31, 39, 49, 59;
15, 21, 28, 36, 45, 55, 66, 78;
19, 25, 33, 41, 51, 62, 73, 86, 99;
23, 30, 38, 47, 57, 69, 81, 94, 108, 123;
...
(End)
MATHEMATICA
PROG
(PARI)
\\ This code produces the triangle mentioned in the example section
r = (3 +sqrt(5))/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) = v[i] + u[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 19 2017
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Mar 18 2017
STATUS
approved