A283943 Interspersion of the signature sequence of e (a rectangular array, by antidiagonals).

1, 4, 2, 10, 6, 3, 19, 13, 8, 5, 30, 23, 16, 11, 7, 44, 35, 27, 20, 14, 9, 61, 50, 40, 32, 24, 17, 12, 81, 68, 56, 46, 37, 28, 21, 15, 103, 89, 75, 63, 52, 42, 33, 25, 18, 128, 112, 97, 83, 70, 58, 48, 38, 29, 22, 156, 138, 121, 106, 91, 77, 65, 54, 43, 34

Offset

1,2

Comments

Row n is the ordered sequence of numbers k such that A023123(k) = n. As a sequence, A283943 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  30  44  61  81   103
2  6   13  23  35  50  68  89   112
3  8   16  27  40  56  75  97   121
5  11  20  32  46  63  83  106  131
7  14  23  37  52  70  91  115  141
9  17  28  42  58  77  99  124  151

Mathematica

r = E; 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}] (* A022786, col 1 of A283943 *)
v = Table[s[n], {n, 0, z}] (* A022785, row 1 of A283943*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (*A 283943, array*)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283943, sequence*)

Prog

(PARI)
\\ Produces the triangle when the array is read by antidiagonals
r = exp(1);
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*e))
def p(n): return n + 1 + sum([int(math.floor((n - k)/e)) 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. A001113, A023123, A022786, A022785.

Sequence in context: A198326 A309989 A283938 * A283942 A266418 A193422

Adjacent sequences: A283940 A283941 A283942 * A283944 A283945 A283946

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Mar 26 2017

Status

approved