A014192 Palindromes in base 4 (written in base 10).

0, 1, 2, 3, 5, 10, 15, 17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255, 257, 273, 289, 305, 325, 341, 357, 373, 393, 409, 425, 441, 461, 477, 493, 509, 514, 530, 546, 562, 582, 598, 614, 630, 650, 666

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

Rajasekaran, Shallit, & Smith prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Patrick De Geest, Palindromic numbers beyond base 10.
• Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
• Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of palindromes: an approach via automata, arXiv:1706.10206 [cs.FL], 2017.
• Index entries for sequences that are an additive basis, order 3.

Crossrefs

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

Programs

Magma

[n: n in [0..800] | Intseq(n, 4) eq Reverse(Intseq(n, 4))]; // Vincenzo Librandi, Sep 09 2015

Mathematica

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,4], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal4Q[n_]:=Module[{c=IntegerDigits[n,4]},c==Reverse[c]]; Select[Range[ 0,700],pal4Q] (* Harvey P. Dale, Jul 21 2020 *)

PARI

ispal(n,b=4)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

Python

from gmpy2 import digits
def A014192(n):
    if n == 1: return 0
    y = (x:=1<<(n.bit_length()-2&-2))<<2
    return (c:=n-x)*x+int(digits(c,4)[-2::-1]or'0',4) if nChai Wah Wu, Jun 14 2024

Formula

Sum_{n>=2} 1/a(n) = 2.7857715... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

Extensions

More terms from Patrick De Geest