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A054238 Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j. 28
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Wikipedia, Z-order Curve
FORMULA
T(n,k) = A000695(k) + 2*A000695(n). - Philippe Deléham, Oct 18 2011
From Robert Israel, Jul 21 2016: (Start)
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)
EXAMPLE
From Philippe Deléham, Oct 18 2011: (Start)
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]
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A000695 (row n=0), A062880 (column k=0), A001196 (main diagonal).
Cf. A059905, A059906, A346453 (by upwards antidiagonals).
See also A163357 and A163334 for other fractal curves in N x N grids.
Sequence in context: A269375 A135141 A098709 * A225589 A245603 A371591
KEYWORD
easy,nonn,base,tabl
AUTHOR
Marc LeBrun, Feb 07 2000
STATUS
approved

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Last modified July 13 22:24 EDT 2024. Contains 374288 sequences. (Running on oeis4.)