A163325 -id:A163325 - cp's OEIS frontend

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 5, 4, 3, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 9, 10, 11, 11, 10, 9, 9, 10, 11, 12, 13, 14, 14, 13, 12

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163528
Index entries for sequences related to coordinates of 2D curves

Cf. A025581, A163335, A002262, A163337, A163325, A163332.

See also: A059252, A059253, A163529, A163530, A163531, A163532.

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

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

Original entry on oeis.org

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

A163326 Pick digits at the odd distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 3, 3, 3, 4, 4, 4, 5, 5, 5, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 6, 6, 6, 7, 7, 7, 8, 8, 8, 6, 6, 6, 7, 7, 7, 8, 8, 8, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0

Offset: 0

Antti Karttunen, Jul 29 2009

			42 in ternary base (A007089) is written as '1120' (1*27 + 1*9 + 2*3 + 0), from which we pick the first and 3rd digits from the right (zero-based!), giving '12' = 1*3 + 2 = 5, thus a(42) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..728
Kevin Ryde, Plot2 of X=A163325,Y=A163326, illustrating the ternary Z-order curve.
Index entries for sequences related to coordinates of 2D curves

A059906 is an analogous sequence for binary. Note that A037314(A163325(n)) + 3*A037314(A163326(n)) = n for all n. Cf. A007089, A163327-A163329.

a(n) = fromdigits(digits(n,9)\3,3); \\ Kevin Ryde, May 15 2020

a(n) = A163325(floor(n/3))

a(n) = Sum_{k>=0} A030341(n,k)*b(k) with (b) = (0,1,0,3,0,9,0,27,0,81,0,243,0,...): powers of 3 alternating with zeros. - Philippe Deléham, Oct 22 2011

Edited by Charles R Greathouse IV, Nov 01 2009

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.

A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

Original entry on oeis.org

0, 4, 8, 36, 40, 44, 72, 76, 80, 324, 328, 332, 360, 364, 368, 396, 400, 404, 648, 652, 656, 684, 688, 692, 720, 724, 728, 2916, 2920, 2924, 2952, 2956, 2960, 2988, 2992, 2996, 3240, 3244, 3248, 3276, 3280, 3284, 3312, 3316, 3320, 3564, 3568, 3572, 3600, 3604

Offset: 0

Kevin Ryde, Oct 09 2020

Also, numbers whose ternary digit runs are all even lengths (including 0 reckoned as no digits at all).  Also, change ternary digits 0,1,2 to base 9 digits 0,4,8, and hence numbers which can be written in base 9 using only digits 0,4,8.
Digit duplication 00,11,22 can be compared to A037314 which is 0 above each so 00,01,02, or A208665 which is 0 below each so 00,10,20.  Duplication is the sum of these, or any one is a suitable multiple of another (*3, *4, etc).
This sequence is the points on the X=Y diagonal of the ternary Z-order curve (see example table in A163328).  The Z-order curve takes a point number p and splits its ternary digits alternately to X and Y coordinates so X(p) = A163325(p) and Y(p) = A163326(p).  Duplicate digits in a(n) are the diagonal X(a(n)) = Y(a(n)) = n.

			n=73 is ternary 2201 which duplicates to 22220011 ternary = 8804 base 9 = 6484 decimal.

Kevin Ryde, Table of n, a(n) for n = 0..6560

Cf. A037314, A208665, A163325, A163326, A163328.
Cf. A020331 (ternary concatenation).
Digit duplication in other bases: A001196, A338754.

PARI
```
a(n) = fromdigits(digits(n,3),9)<<2;
```

from gmpy2 import digits
def A338086(n): return int(''.join(d*2 for d in digits(n,3)),3) # Chai Wah Wu, May 07 2022

a(n) = A037314(n) + A208665(n) = 4*A037314(n) = (4/3)*A208665(n).

a(n) = 4*Sum_{i=0..k} d[i]*9^i where the ternary expansion of n is n = Sum_{i=0..k} d[i]*3^i with digits d[i]=0,1,2.

A163331 Inverse permutation to A163330.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 29 2009

Antti Karttunen, Table of n, a(n) for n = 0..6560
Index entries for sequences that are permutations of the natural numbers

Inverse: A163330. a(n) = A163329(A163327(n)). Cf. A007089.

(define (A163331 n) (A001477bi (A163326 n) (A163325 n)))
(define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

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

A105186 Replace odd-positioned digits with 0 in ternary representation of n.

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 0, 1, 2, 9, 10, 11, 9, 10, 11, 9, 10, 11, 18, 19, 20, 18, 19, 20, 18, 19, 20, 0, 1, 2, 0, 1, 2, 0, 1, 2, 9, 10, 11, 9, 10, 11, 9, 10, 11, 18, 19, 20, 18, 19, 20, 18, 19, 20, 0, 1, 2, 0, 1, 2, 0, 1, 2, 9, 10, 11, 9, 10, 11, 9, 10, 11, 18, 19, 20, 18, 19, 20, 18, 19, 20, 81, 82

Offset: 0

Reinhard Zumkeller, Apr 11 2005

			n = 123 = '11120' --> '10100' = 90 = a(123).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007089, A063694, A163325.

a105186 0 = 0
a105186 n = 9 * a105186 n' + mod t 3
            where (n', t) = divMod n 9
-- Reinhard Zumkeller, Sep 26 2015

a(n) = fromdigits(digits(n,9)%3,9); \\ Kevin Ryde, May 20 2020

a(n) = n - a(floor(n/3))*3, a(0) = 0.

a(n) = 9*a(floor(n/9)) + (n mod 9) mod 3. - Reinhard Zumkeller, Sep 26 2015

cp's OEIS Frontend

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

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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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A163326 Pick digits at the odd distance from the least significant end of the ternary expansion of n, then convert back to decimal.

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A163329 Inverse permutation to A163328.

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A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

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A163331 Inverse permutation to A163330.

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A105186 Replace odd-positioned digits with 0 in ternary representation of n.

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