A283942 Rectangular array by antidiagonals: interspersion of the signature sequence of sqrt(8).

1, 4, 2, 10, 6, 3, 19, 13, 8, 5, 31, 23, 16, 11, 7, 46, 36, 27, 20, 14, 9, 63, 52, 41, 32, 24, 17, 12, 83, 70, 58, 47, 37, 28, 21, 15, 106, 91, 77, 65, 53, 42, 33, 25, 18, 132, 115, 99, 85, 72, 59, 48, 38, 29, 22, 161, 142, 124, 108, 93, 79, 66, 54, 43, 34

Offset

1,2

Comments

Row n is the ordered sequence of numbers k such that A023131(k) = n. As a sequence, A283942 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   4   10   19   31   46   63   83   106
2   6   13   23   36   52   70   91   115
3   8   16   27   41   58   77   99   124
5   11  20   32   47   65   85   108  134
7   14  24   37   53   72   93   117  144
9   17  28   42   59   79   101  126  154
12  21  33   48   66   87   110  136  165

Mathematica

r = Sqrt[8]; 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}] (* A022794 , col 1 of A283942 *)
v = Table[s[n], {n, 0, z}] (* A022793, row 1 of A283942*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283942, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283942, sequence *)

Prog

(PARI)
\\ Produces the triangle when the array is read by antidiagonals
r = sqrt(8);
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 sympy import sqrt
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*sqrt(8)))
def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(8))) 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

Cf. A010466, A023131, A022794, A022793.

Sequence in context: A309989 A283938 A283943 * A266418 A193422 A357595

Adjacent sequences: A283939 A283940 A283941 * A283943 A283944 A283945

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Mar 26 2017

Status

approved