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A054238
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Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j.
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28
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0, 1, 2, 4, 3, 8, 5, 6, 9, 10, 16, 7, 12, 11, 32, 17, 18, 13, 14, 33, 34, 20, 19, 24, 15, 36, 35, 40, 21, 22, 25, 26, 37, 38, 41, 42, 64, 23, 28, 27, 48, 39, 44, 43, 128, 65, 66, 29, 30, 49, 50, 45, 46, 129, 130, 68, 67, 72, 31, 52, 51, 56, 47, 132, 131, 136, 69, 70, 73, 74
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OFFSET
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0,3
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COMMENTS
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Inverse of sequence A054239 considered as a permutation of the nonnegative integers.
Permutation of nonnegative integers. Can be used as natural alternate number casting for pairs/tables (vs. usual diagonalization).
This array is a Z-order curve in an N x N grid. - Max Barrentine, Sep 24 2015
Each row n of this array is the lexicographically earliest sequence such that no term occurs in a previous row, no three terms form an arithmetic progression, and the k-th term in the n-th row is equal to the k-th term in row 0 plus some constant (specifically, T(n,k) = T(0,k) + A062880(n)). - Max Barrentine, Jul 20 2016
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LINKS
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FORMULA
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G.f. of array: g(x,y) = (1/(1-x)*(1-y)) * Sum_{i>=0}
(2^(2*i+1)*x^(2^i)/(1+x^(2^i)) + 2^(2*i)*y^(2^i)/(1+y^(2^i))).
T(2*n+i,2*k+j) = 4*T(n,k) + 2*i+j for i,j in {0,1}. (End)
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EXAMPLE
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The array starts in row n=0 with columns k >= 0 as follows:
0 1 4 5 16 17 20 21 ...
2 3 6 7 18 19 22 23 ...
8 9 12 13 24 25 28 29 ...
10 11 14 15 26 27 30 31 ...
32 33 36 37 48 49 52 53 ...
34 35 38 39 50 51 54 55 ...
40 41 44 45 56 57 60 61 ...
42 43 46 47 58 59 62 63 ...
(End)
T(6,5)=57 because 1.1.0. (6) merged with .1.0.1 (5) is 111001 (57). [Corrected by Georg Fischer, Jan 21 2022]
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MAPLE
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N:= 4: # to get the first 2^(2N+1)+2^N terms
G:= 1/(1-y)/(1-x)*(add(2^(2*i+1)*x^(2^i)/(1+x^(2^i)), i=0..N) + add(2^(2*i)*y^(2^i)/(1+y^(2^i)), i=0..N)):
S:= mtaylor(G, [x=0, y=0], 2^(N+1)):
seq(seq(coeff(coeff(S, x, i), y, m-i), i=0..m), m=0..2^(N+1)-1); # Robert Israel, Jul 21 2016
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MATHEMATICA
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Table[Total@ Map[FromDigits[#, 2] &, Insert[#, 0, {-1, -1}] &@ Map[Riffle[IntegerDigits[#, 2], 0, 2] &, {n - k, k}]], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)
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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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