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

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

Author

Kevin Ryde, Oct 09 2020

Keywords

• base
• nonn

Comments

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.

Examples

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

Links

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

Crossrefs

Cf. A037314, A208665, A163325, A163326, A163328.

Cf. A020331 (ternary concatenation).

Digit duplication in other bases: A001196, A338754.

Programs

PARI

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

Python

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

Formula

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.