A057300 -id:A057300 - cp's OEIS frontend

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

0, 3, 6, 1, 4, 7, 2, 5, 8, 27, 30, 33, 28, 31, 34, 29, 32, 35, 54, 57, 60, 55, 58, 61, 56, 59, 62, 9, 12, 15, 10, 13, 16, 11, 14, 17, 36, 39, 42, 37, 40, 43, 38, 41, 44, 63, 66, 69, 64, 67, 70, 65, 68, 71, 18, 21, 24, 19, 22, 25, 20, 23, 26, 45, 48, 51, 46, 49, 52, 47, 50, 53

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '(000...)102' (... + 0*27 + 1*9 + 0*3 + 2), which results '1020' = 1*27 + 0*9 + 2*3 + 0 = 33, when the odd- and even-positioned digits are swapped, thus a(11) = 33.

Antti Karttunen, Table of n, a(n) for n = 0..728
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A163328-A163329.

Other bases: A057300, A126006, A217558.

from sympy.ntheory import digits
def a(n):
    d = digits(n, 3)[1:]
    return sum(3**(i+(1-2*(i&1)))*di for i, di in enumerate(d[::-1]))
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 05 2022

(define (A163327 n) (+ (A037314 (A163326 n)) (* 3 (A037314 (A163325 n)))))

a(n) = A037314(A163326(n)) + 3*A037314(A163325(n))

Edited by Charles R Greathouse IV, Nov 01 2009

A163233 Two-dimensional Binary Reflected Gray Code: a(i,j) = bits of binary expansion of A003188(i) interleaved with that of A003188(j).

Original entry on oeis.org

0, 1, 2, 5, 3, 10, 4, 7, 11, 8, 20, 6, 15, 9, 40, 21, 22, 14, 13, 41, 42, 17, 23, 30, 12, 45, 43, 34, 16, 19, 31, 28, 44, 47, 35, 32, 80, 18, 27, 29, 60, 46, 39, 33, 160, 81, 82, 26, 25, 61, 62, 38, 37, 161, 162, 85, 83, 90, 24, 57, 63, 54, 36, 165, 163, 170, 84, 87, 91

Offset: 0

Antti Karttunen, Jul 29 2009

The top left 8 X 8 corner of the array is
+0 +1 +5 +4 20 21 17 16
+2 +3 +7 +6 22 23 19 18
10 11 15 14 30 31 27 26
+8 +9 13 12 28 29 25 24
40 41 45 44 60 61 57 56
42 43 47 46 62 63 59 58
34 35 39 38 54 55 51 50
32 33 37 36 52 53 49 48
By taking the top left 2 X 2 corner, 2 X 4 rectangle ((0,1,5,4),(2,3,7,6)) or 4 X 4 corner one obtains Karnaugh map templates for 2, 3 or 4 variables respectively (although not the standard ones usually given in the textbooks).

Antti Karttunen, Table of n, a(n) for n = 0..2079
Wikipedia, Karnaugh map
Index entries for sequences that are permutations of the natural numbers

Inverse: A163234. a(n) = A057300(A163235(n)). Transpose: A163235. Row sums: A163242. Cf. A054238, A147995.

Table[Function[k, FromDigits[#, 2] &@ Apply[Function[{a, b}, Riffle @@ Map[PadLeft[#, Max[Length /@ {a, b}]] &, {a, b}]], Map[IntegerDigits[#, 2] &@ BitXor[#, Floor[#/2]] &, {k, j}]]][i - j], {i, 0, 11}, {j, i, 0, -1}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

def a000695(n):
    n=bin(n)[2:]
    x=len(n)
    return sum([int(n[i])*4**(x - 1 - i) for i in range(x)])
def a003188(n): return n^(n>>1)
def a(n, k): return a000695(a003188(n)) + 2*a000695(a003188(k))
for n in range(21): print([a(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 25 2017

(define (A163233bi x y) (+ (A000695 (A003188 x)) (* 2 (A000695 (A003188 y)))))
(define (A163233 n) (A163233bi (A025581 n) (A002262 n)))

a(x,y) = A000695(A003188(x)) + 2*A000695(A003188(y)).

A163235 Two-dimensional Binary Reflected Gray Code, transposed version: a(i,j) = bits of binary expansion of A003188(j) interleaved with that of A003188(i).

Original entry on oeis.org

0, 2, 1, 10, 3, 5, 8, 11, 7, 4, 40, 9, 15, 6, 20, 42, 41, 13, 14, 22, 21, 34, 43, 45, 12, 30, 23, 17, 32, 35, 47, 44, 28, 31, 19, 16, 160, 33, 39, 46, 60, 29, 27, 18, 80, 162, 161, 37, 38, 62, 61, 25, 26, 82, 81, 170, 163, 165, 36, 54, 63, 57, 24, 90, 83, 85, 168, 171, 167

Offset: 0

Antti Karttunen, Jul 29 2009

The top left 8x8 corner of this array
+0 +2 10 +8 40 42 34 32
+1 +3 11 +9 41 43 35 33
+5 +7 15 13 45 47 39 37
+4 +6 14 12 44 46 38 36
20 22 30 28 60 62 54 52
21 23 31 29 61 63 55 53
17 19 27 25 57 59 51 49
16 18 26 24 56 58 50 48
corresponds with Adamson's "H-bond codon-anticodon magic square" (see page 287 in Pickover's book):
CCC CCU CUU CUC UUC UUU UCU UCC
CCA CCG CUG CUA UUA UUG UCG UCA
CAA CAG CGG CGA UGA UGG UAG UAA
CAC CAU CGU CGC UGC UGU UAU UAC
AAC AAU AGU AGC GGC GGU GAU GAC
AAA AAG AGG AGA GGA GGG GAG GAA
ACA ACG AUG AUA GUA GUG GCG GCA
ACC ACU AUU AUC GUC GUU GCU GCC
when the base-triples are interpreted as quaternary (base-4) numbers, with the following rules: C = 0, A = 1, U = 2, G = 3.

Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002, pp. 285-289.

Antti Karttunen, Table of n, a(n) for n = 0..2079
Index entries for sequences that are permutations of the natural numbers

Inverse: A163236. a(n) = A057300(A163233(n)). Transpose: A163233. Row sums: A163242. Cf. A054238, A147995.

Table[Function[k, FromDigits[#, 2] &@Apply[Function[{a, b}, Riffle @@ Map[PadLeft[#, Max[Length /@ {a, b}]] &, {a, b}]], Map[IntegerDigits[#, 2] &@ BitXor[#, Floor[#/2]] &, {k, j}]]][i - j], {i, 0, 11}, {j, 0, i}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

def a000695(n):
    n=bin(n)[2:]
    x=len(n)
    return sum(int(n[i])*4**(x - 1 - i) for i in range(x))
def a003188(n): return n^(n>>1)
def a(n, k): return a000695(a003188(n)) + 2*a000695(a003188(k))
for n in range(21): print([a(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 25 2017

(define (A163235 n) (A163233bi (A002262 n) (A025581 n)))

A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).

Original entry on oeis.org

0, 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 64, 68, 72, 76, 65, 69, 73, 77, 66, 70, 74, 78, 67, 71, 75, 79, 128, 132, 136, 140, 129, 133, 137, 141, 130, 134, 138, 142, 131, 135, 139, 143, 192, 196, 200, 204, 193, 197, 201, 205, 194, 198, 202, 206, 195

Offset: 0

Antti Karttunen, Jan 02 2007

			29 = 0*64 + 1*16 + 3*4 + 1, i.e., 131 in quaternary and when digits are swapped in pairs, results 1013 in quaternary (1*64 + 0*16 + 1*4 + 3 = 71 in decimal), thus a(29)=71.

Index entries for sequences that are permutations of the natural numbers

Cf. A126007. A057300 is the analogous permutation based on swapping the binary digits of n.

Cf. A004442.

#include 
uint32_t a(uint32_t n) { return ((n & 0x33333333) << 2) | ((n & 0xcccccccc) >> 2); } /* Falk Hüffner, Jan 23 2022 */

a(n) = my(d=Vecrev(digits(n, 4))); if (#d % 2, d = concat(d, 0)); fromdigits(Vecrev(vector(#d, i, d[i+(-1)^(i-1)])), 4); \\ Michel Marcus, Jan 23 2022

(define (A126006 n) (let loop ((n n) (s 0) (p 1)) (cond ((zero? n) s) (else (loop (floor->exact (/ n 16)) (+ s (* p (+ (* 4 (modulo n 4)) (modulo (floor->exact (/ n 4)) 4)))) (* p 16))))))

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 14, 12, 13, 5, 6, 4, 7, 10, 9, 11, 8, 21, 20, 22, 23, 16, 19, 17, 18, 26, 27, 25, 24, 31, 28, 30, 29, 63, 62, 60, 61, 48, 49, 51, 50, 58, 57, 59, 56, 53, 54, 52, 55, 42, 43, 41, 40, 47, 44, 46, 45, 37, 36, 38, 39, 32, 35, 33, 34, 255, 254, 252, 253, 240, 241, 243, 242, 250, 249, 251, 248, 245, 246, 244

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302846 (inverse).

Cf. A006068, A057300, A064707, A163355, A302843, A302782.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A064707(n) = A006068(A006068(n));
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302845(n) = A163355(A064707(n));

a(n) = A163355(A064707(n)).

a(n) = A302843(A006068(n)).

A106485 CGT-tree negating involution of nonnegative integers.

Original entry on oeis.org

0, 2, 1, 3, 32, 34, 33, 35, 16, 18, 17, 19, 48, 50, 49, 51, 8, 10, 9, 11, 40, 42, 41, 43, 24, 26, 25, 27, 56, 58, 57, 59, 4, 6, 5, 7, 36, 38, 37, 39, 20, 22, 21, 23, 52, 54, 53, 55, 12, 14, 13, 15, 44, 46, 45, 47, 28, 30, 29, 31, 60, 62, 61, 63, 128, 130, 129, 131, 160, 162

Offset: 0

Antti Karttunen, May 21 2005

nonn

This involution negates game trees used in the combinatorial game theory, when they are encoded in the way explained in A106486.

Cycles are confined into ranges [a(n),a(n+1)[, where a(0)=0 and a(n+1)=2^(2*a(n)), i.e. the ranges are [0,0], [1,3], [4,255], [256,(2^512)-1], ...

Antti Karttunen, Table of n, a(n) for n = 0..1023
Index entries for sequences that are permutations of the natural numbers

A057300 is a "shallow" version which just swaps the left and right options of the game tree, but does not reflect the subtrees themselves. Cf. A106486-A106487.

(define (A106485 n) (let loop ((n n) (i 0) (s 0)) (cond ((zero? n) s) ((odd? n) (loop (/ (- n 1) 2) (1+ i) (+ s (if (even? i) (expt 2 (+ 1 (* 2 (A106485 (/ i 2))))) (expt 2 (* 2 (A106485 (/ (- i 1) 2)))))))) (else (loop (/ n 2) (1+ i) s)))))

A163485 Permutation of integers used for constructing A147995 and A163545.

Original entry on oeis.org

0, 3, 1, 2, 14, 15, 13, 12, 6, 7, 5, 4, 8, 11, 9, 10, 58, 57, 59, 56, 62, 63, 61, 60, 54, 55, 53, 52, 50, 49, 51, 48, 26, 25, 27, 24, 30, 31, 29, 28, 22, 23, 21, 20, 18, 17, 19, 16, 32, 35, 33, 34, 46, 47, 45, 44, 38, 39, 37, 36, 40, 43, 41, 42, 234, 233, 235, 232, 228, 229

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

Inverse: A163486. This permutation can be used to construct array A147995 and its transpose A163545. See A163355 for a bit similarly defined recursive permutation.

A302843 Permutation of nonnegative integers: a(n) = A163355(A006068(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 13, 14, 15, 10, 9, 8, 11, 4, 7, 6, 5, 26, 27, 24, 25, 30, 29, 28, 31, 16, 19, 18, 17, 22, 23, 20, 21, 42, 43, 40, 41, 46, 45, 44, 47, 32, 35, 34, 33, 38, 39, 36, 37, 58, 57, 56, 59, 52, 55, 54, 53, 48, 49, 50, 51, 60, 61, 62, 63, 192, 193, 194, 195, 204, 205, 206, 207, 202, 201, 200, 203, 196, 199, 198, 197, 218, 219, 216, 217, 222

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302844 (inverse).

Cf. A003188, A006068, A057300, A163355, A302845.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A057300(n) = { my(t=1, s=0); while(n>0,  if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302843(n) = A163355(A006068(n));

a(n) = A163355(A006068(n)).

a(n) = A302845(A003188(n)).

A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 15, 14, 13, 10, 9, 8, 11, 4, 5, 6, 7, 24, 27, 26, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 21, 20, 23, 40, 43, 42, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 37, 36, 39, 56, 57, 58, 59, 52, 55, 54, 53, 50, 49, 48, 51, 60, 61, 62, 63, 192, 195, 194, 193, 198, 199, 196, 197, 202, 203, 200, 201, 206, 205

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

When A207901, which is a multiplicative walk permutation, is composed from the right with this permutation, the result is A302781, another multiplicative walk permutation.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302843 (inverse).

Cf. A003188, A006068, A057300, A163356, A302846, A302781.

A003188(n) = bitxor(n, n>>1);
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r)))); \\ Antti Karttunen, Apr 14 2018
A302844(n) = A003188(A163356(n));

a(n) = A003188(A163356(n)).

a(n) = A006068(A302846(n)).

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)

cp's OEIS Frontend

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

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A163233 Two-dimensional Binary Reflected Gray Code: a(i,j) = bits of binary expansion of A003188(i) interleaved with that of A003188(j).

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A163235 Two-dimensional Binary Reflected Gray Code, transposed version: a(i,j) = bits of binary expansion of A003188(j) interleaved with that of A003188(i).

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A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4*256 + q3*64 + q2*16 + q1*4 + q0).

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A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

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A106485 CGT-tree negating involution of nonnegative integers.

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A163485 Permutation of integers used for constructing A147995 and A163545.

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A302843 Permutation of nonnegative integers: a(n) = A163355(A006068(n)).

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A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).