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

1,3

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.

Formula

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

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 *)

Prog

(Magma) [n: n in [0..800] | Intseq(n, 4) eq Reverse(Intseq(n, 4))]; // Vincenzo Librandi, Sep 09 2015
(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 n<x+y else (c:=n-y)*y+int(digits(c, 4)[-1::-1]or'0', 4) # Chai Wah Wu, Jun 14 2024

Crossrefs

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

Sequence in context: A043707 A296694 A297253 * A250746 A048315 A093832

Adjacent sequences: A014189 A014190 A014191 * A014193 A014194 A014195

Keyword

nonn,base,easy

Author

N. J. A. Sloane

Extensions

More terms from Patrick De Geest

Status

approved