A054239 -id:A054239 - cp's OEIS frontend

A054238 Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j.

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

Offset: 0

Marc LeBrun, Feb 07 2000

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

			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]

Robert Israel, Table of n, a(n) for n = 0..10000
G. M. Morton, A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, IBM, 1966.
Wikipedia, Z-order Curve
Index entries for sequences that are permutations of the natural numbers

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.

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

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 *)

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)

A163358 Inverse permutation to A163357.

Original entry on oeis.org

0, 1, 4, 2, 5, 9, 13, 8, 12, 18, 24, 17, 11, 7, 3, 6, 10, 16, 22, 15, 21, 28, 37, 29, 38, 47, 58, 48, 39, 30, 23, 31, 40, 50, 60, 49, 59, 70, 83, 71, 84, 97, 112, 98, 85, 72, 61, 73, 62, 52, 42, 51, 41, 32, 25, 33, 26, 19, 14, 20, 27, 34, 43, 35, 44, 54, 64, 53, 63, 74, 87

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

abs(A025581(a(n+1)) - A025581(a(n))) + abs(A002262(a(n+1)) - A002262(a(n))) = 1 for all n.

Inverse: A163357. a(n) = A054239(A163356(n)). One-based version: A163362. See also A163334 and A163336.

A163329 Inverse permutation to A163328.

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 5, 8, 12, 6, 10, 15, 11, 16, 22, 17, 23, 30, 21, 28, 36, 29, 37, 46, 38, 47, 57, 9, 13, 18, 14, 19, 25, 20, 26, 33, 24, 31, 39, 32, 40, 49, 41, 50, 60, 48, 58, 69, 59, 70, 82, 71, 83, 96, 27, 34, 42, 35, 43, 52, 44, 53, 63, 51, 61, 72, 62, 73, 85, 74, 86, 99

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Inverse: A163328. a(n) = A163331(A163327(n)). A054239 is an analogous sequence for binary. Cf. A007089.

(define (A163329 n) (A001477bi (A163325 n) (A163326 n)))
(define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

a(n) = A001477bi(A163325(n),A163326(n)), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 8, 6, 4, 5, 5, 9, 9, 5, 5, 6, 16, 6, 12, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 12, 18, 16, 18, 12, 18, 8, 9, 9, 13, 13, 17, 17, 13, 13, 9, 9, 10, 12, 10, 24, 18, 20, 18, 24, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11, 12, 14, 32, 14

Offset: 0

Marc LeBrun, Feb 07 2000

			T(3,1)=6 because (0*2 + 1*sqrt(2) + 1*1) + (0*2 + 0*sqrt(2) + 1*1) = (1*2 + 1*sqrt(2) + 0*1) (i.e., base sqrt(2) addition).

Reinhard Zumkeller, Antidiagonals n=0..125 of square array, flattened

Cf. A000695, A054239, A057300, A062880, A352909 (pairs (i,j) such that A(i,j) = i+j).

Cf. A201651 (triangle read by rows).

import Data.Bits (xor, (.&.), shift)
a054240 :: Integer -> Integer -> Integer
a054240 x 0 = x
a054240 x y = a054240 (x `xor` y) (shift (x .&. y) 2)
a054240_adiag n =  map (\k -> a054240 (n - k) k) [0..n]
a054240_square = map a054240_adiag [0..]
-- Reinhard Zumkeller, Dec 03 2011

From Peter Munn, Dec 10 2019: (Start)
A(m,0) = A(0,m) = m.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = 4*m.
A(2*n, 2*k) = 2*A(n,k).
A(A000695(n), A000695(k)) = A000695(n+k).
A(A000695(n), 2*A000695(k)) = A000695(n) + 2*A000695(k).
A(A000695(n) + 2*A000695(m), k) = A(A000695(n), k) + A(2*A000695(m), k) - k.
A(A057300(n), A057300(k)) = A057300(A(n,k)).
(End)

A163546 Inverse permutation to A163545.

Original entry on oeis.org

0, 2, 4, 1, 13, 9, 5, 8, 12, 18, 24, 17, 11, 7, 3, 6, 62, 51, 42, 52, 43, 35, 27, 34, 26, 19, 14, 20, 41, 33, 25, 32, 40, 50, 60, 49, 85, 73, 61, 72, 84, 98, 112, 97, 83, 71, 59, 70, 58, 47, 38, 48, 39, 31, 23, 30, 22, 15, 10, 16, 37, 29, 21, 28, 268, 245, 224, 246, 183

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

Inverse: A163545. a(n) = A054239(A163486(n)) = A061579(A163544(n)).

A163908 Inverse permutation to A163907.

Original entry on oeis.org

0, 1, 2, 4, 12, 24, 17, 18, 11, 3, 6, 7, 8, 13, 5, 9, 10, 22, 15, 16, 21, 28, 29, 37, 39, 30, 31, 23, 48, 47, 38, 58, 62, 42, 51, 52, 41, 32, 33, 25, 27, 34, 35, 43, 20, 19, 26, 14, 73, 61, 85, 72, 71, 70, 59, 83, 49, 50, 40, 60, 84, 97, 98, 112, 144, 180, 161, 162, 179

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Inverse: A163907. a(n) = A054239(A163906(n)) = A163358(A163356(n)). See also A163358, A163918.

A163918 Inverse permutation to A163917.

Original entry on oeis.org

0, 1, 4, 2, 11, 6, 7, 3, 8, 5, 9, 13, 18, 17, 12, 24, 10, 15, 16, 22, 21, 28, 37, 29, 48, 47, 58, 38, 23, 30, 39, 31, 73, 85, 72, 61, 71, 70, 83, 59, 84, 97, 112, 98, 60, 50, 49, 40, 14, 26, 20, 19, 25, 32, 41, 33, 52, 42, 62, 51, 27, 34, 43, 35, 260, 237, 238, 216, 215

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Inverse: A163917. a(n) = A054239(A163916(n)) = A163908(A163356(n)). See also A163358, A163908.

A072794 Inverse permutation to A072793.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 8, 9, 12, 11, 14, 15, 16, 21, 13, 23, 17, 18, 36, 19, 20, 25, 22, 27, 28, 29, 40, 30, 31, 38, 24, 44, 45, 46, 55, 26, 57, 32, 33, 78, 34, 35, 42, 37, 82, 49, 50, 59, 39, 61, 47, 48, 136, 53, 54, 63, 56, 65, 66, 67, 86, 68, 69, 80, 41, 90, 91, 92

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Composition of A054239 & A048679.

A054241 Number of partitions of bit-interleaved numbers; partitions of n in base sqrt(2).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 2, 4, 3, 7, 9, 16, 16, 31, 5, 7, 12, 19, 11, 15, 30, 45, 29, 47, 57, 97, 77, 118, 162, 257, 5, 12, 7, 19, 29, 57, 47, 97, 11, 30, 15, 45, 77, 162, 118, 257, 109, 189, 189, 339, 323, 522, 589, 975, 323, 589, 522, 975, 1043, 1752, 1752

Offset: 0

Marc LeBrun, Feb 07 2000

Rearrangement of A054225 via A054238. Also can be directly derived from A054240 (bit-interleaved addition table).

			a(6)=4 thus: {6, 4+2, 3+1, 2+1+1} all in base sqrt(2).
From _Sean A. Irvine_, Jan 26 2022: (Start)
a(12)=9 from {12, 9+1, 8+4, 8+1+1, 6+2, 4+2+2, 3+3, 3+2+1, 2+2+1+1}.
a(13)=16 from {13, 12+1, 9+4, 9+1+1, 8+5, 8+4+1, 8+1+1+1, 7+2, 6+3, 6+2+1, 5+2+2, 4+3+2, 4+2+2+1, 3+3+1, 3+2+1+1, 2+2+1+1+1}.
(End)

Cf. A054225, A054238, A054239, A054240.

a(n) = A054225(A054239(n)). - Sean A. Irvine, Jan 26 2022

Data corrected by Sean A. Irvine, Jan 26 2022

cp's OEIS Frontend

A054238 Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j.

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A163358 Inverse permutation to A163357.

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A163329 Inverse permutation to A163328.

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A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

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A163546 Inverse permutation to A163545.

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A163908 Inverse permutation to A163907.

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A163918 Inverse permutation to A163917.

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A072794 Inverse permutation to A072793.

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A054241 Number of partitions of bit-interleaved numbers; partitions of n in base sqrt(2).

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