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A283943
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Interspersion of the signature sequence of e (a rectangular array, by antidiagonals).
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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*)
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PROG
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(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(); ); };
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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