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A283939
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Interspersion of the signature sequence of sqrt(2).
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2
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1, 3, 2, 6, 5, 4, 11, 9, 8, 7, 17, 15, 13, 12, 10, 25, 22, 20, 18, 16, 14, 34, 31, 28, 26, 23, 21, 19, 44, 41, 38, 35, 32, 29, 27, 24, 56, 52, 49, 46, 42, 39, 36, 33, 30, 69, 65, 61, 58, 54, 50, 47, 43, 40, 37, 84, 79, 75, 71, 67, 63, 59, 55, 51, 48, 45, 100
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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 A007336(k)=n. As a sequence, A283939 is a permutation of the positive integers. As an array, A283939 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = sqrt(2). 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 3 6 11 17 25 34 44 56
2 5 9 15 22 31 41 52 65
4 8 13 20 28 38 49 61 75
7 12 18 26 35 46 58 71 86
10 16 23 32 42 54 67 81 97
14 21 29 39 50 63 77 91 109
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MATHEMATICA
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r = Sqrt[2]; 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}] (* A022776, col 1 of A283939 *)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)
p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)
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PROG
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(PARI)
r = sqrt(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) = 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)
from sympy import sqrt
import math
def s(n): return 1 if n<1 else s(n - 1) + 1 +
int(math.floor(n*sqrt(2)))
def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(2))) 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 21 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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