A023745 Plaindromes: numbers whose digits in base 3 are in nondecreasing order.

0, 1, 2, 4, 5, 8, 13, 14, 17, 26, 40, 41, 44, 53, 80, 121, 122, 125, 134, 161, 242, 364, 365, 368, 377, 404, 485, 728, 1093, 1094, 1097, 1106, 1133, 1214, 1457, 2186, 3280, 3281, 3284, 3293, 3320, 3401, 3644, 4373, 6560, 9841, 9842, 9845, 9854

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• base
• easy

Examples

			In base 3 these numbers are 0, 1, 2, 11, 12, 22, 111, 112, 122, 222, 1111, 1112, ... [corrected by _Sean A. Irvine_, Jun 10 2019]
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• A. V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, arXiv:1809.00122 [math.CA], 2018.

Crossrefs

Cf. A023746 onwards. In base 2 we get A000225.

Programs

Mathematica

Select[Range[0,10000],!Negative[Min[Differences[IntegerDigits[ #,3]]]]&] (* or *) With[{nn=10},FromDigits[#,3]&/@Union[Flatten[Table[ PadRight[ PadLeft[{},n,1],x,2],{n,0,nn},{x,0,nn}],1]]] (* Harvey P. Dale, Oct 12 2011 *)
Select[Range[0,10000],LessEqual@@IntegerDigits[#,3]&] (* Ray Chandler, Jan 06 2014 *)

Python

from math import isqrt
def A023745(n): return (3**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+3**(n-1-(a*(a+1)>>1))>>1)-1 # Chai Wah Wu, Apr 08 2025

Formula

Numbers that in ternary are the concatenation of i 1's with j 2's, i, j>=0. Also a(n) = A073216(n+1) - 1. Proof: Write a(n) as 1{m}2{n}, then adding 1 gives 1{m-1}20{n} for m>0 and 10{n} for m=0. Doubling yields 10{m-1}10{n} or 20{n}, respectively. These two forms exactly describe the forms of sums of two powers of 3, the two powers being 3^n and 3^(m+n). - Hugo van der Sanden

Extensions

Change offset to 1 by Ray Chandler, Jan 06 2014