

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
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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 Zorder 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 kth term in the nth row is equal to the kth term in row 0 plus some constant (specifically, T(n,k) = T(0,k) + A062880(n)).  Max Barrentine, Jul 20 2016


LINKS



FORMULA

G.f. of array: g(x,y) = (1/(1x)*(1y)) * 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

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/(1y)/(1x)*(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, mi), 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



KEYWORD



AUTHOR



STATUS

approved



