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
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.
Programs
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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