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

1,3

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.

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

Example

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]

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A334522 A102829 A031988 * A217136 A294944 A178656

Adjacent sequences: A023742 A023743 A023744 * A023746 A023747 A023748

Keyword

nonn,base,easy

Author

Olivier Gérard

Extensions

Change offset to 1 by Ray Chandler, Jan 06 2014

Status

approved