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A283944
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Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).
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2
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1, 5, 2, 12, 7, 3, 22, 15, 9, 4, 35, 26, 18, 11, 6, 51, 40, 30, 21, 14, 8, 70, 57, 45, 34, 25, 17, 10, 92, 77, 63, 50, 39, 29, 20, 13, 118, 100, 84, 69, 56, 44, 33, 24, 16, 147, 127, 108, 91, 76, 62, 49, 38, 28, 19, 179, 157, 136, 116, 99, 83, 68, 55, 43, 32
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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 A023133(k) = n. As a sequence, it 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 5 12 22 35 51 70 92 118
2 7 15 26 40 57 77 100 127
3 9 18 30 45 63 84 108 136
4 11 21 34 50 69 91 115 145
6 14 25 39 56 76 99 125 155
8 17 29 44 62 83 107 134 165
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MATHEMATICA
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r = Pi; 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}] (* A022796, col 1 of A283944 *)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283944, array*)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283944, sequence *)
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PROG
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(PARI)
\\ Produces the triangle when the array is read by antidiagonals
r = Pi;
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*pi))
def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) 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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