A118594 -id:A118594 - cp's OEIS frontend

A118600 Palindromes in base 9 (written in base 9).

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 22, 33, 44, 55, 66, 77, 88, 101, 111, 121, 131, 141, 151, 161, 171, 181, 202, 212, 222, 232, 242, 252, 262, 272, 282, 303, 313, 323, 333, 343, 353, 363, 373, 383, 404, 414, 424, 434, 444, 454, 464, 474, 484, 505, 515, 525, 535, 545

Offset: 1

Martin Renner, May 08 2006

Chai Wah Wu, Table of n, a(n) for n = 1..1457

Cf. A029955, A057148, A118594, A118595, A118596, A118597, A118598, A118599, A118600, A002113.

(* get NextPalindrome from A029965 *) Select[NestList[NextPalindrome, 0, 62], Max@IntegerDigits@# < 9 &] (* Robert G. Wilson v, May 09 2006 *)

from gmpy2 import digits
def palgenbase(l,b): # generator of palindromes in base b <=10 of length <= 2*l, written in base b
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[-2::-1])
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[::-1])
A118600_list = list(palgenbase(3,9)) # Chai Wah Wu, Dec 01 2014

from sympy import integer_log
from gmpy2 import digits
def A118600(n):
    if n == 1: return 0
    y = 9*(x:=9**integer_log(n>>1,9)[0])
    return int((s:=digits(n-x,9))+s[-2::-1] if nChai Wah Wu, Jun 14 2024

More terms from Robert G. Wilson v, May 09 2006

A111575 Powers of 3 repeated four times.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 9, 9, 9, 9, 27, 27, 27, 27, 81, 81, 81, 81, 243, 243, 243, 243, 729, 729, 729, 729, 2187, 2187, 2187, 2187, 6561, 6561, 6561, 6561, 19683, 19683, 19683, 19683, 59049, 59049, 59049, 59049, 177147, 177147, 177147, 177147, 531441

Offset: 0

Jeremy Gardiner, Nov 17 2005

Generating sequence for the number of 0's and 1's (run lengths) in the parity of A006072, A111065 and A118594.

			a(10) = 3^floor(10/4) = 3^2 = 9.

Vincenzo Librandi, Table of n, a(n) for n = 0..8000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3).

Cf. A006072, A111065, A108411, A118594, A127975.

[3^(Floor(n/4)):n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

With[{pt=3^Range[0,15]},Sort[Join[pt,pt,pt,pt]]] (* Harvey P. Dale, Sep 17 2011 *)
Table[PadRight[{},4,3^n],{n,0,15}]//Flatten (* Harvey P. Dale, Jan 17 2019 *)

a(n)=3^(n\4) \\ Charles R Greathouse IV, Oct 03 2016

a(n) = 3^floor(n/4).

O.g.f.: -(1+x)*(1+x^2)/(-1+3*x^4). - R. J. Mathar, Jan 08 2008

A263608 Palindromes which are base-3 representations of squares.

Original entry on oeis.org

0, 1, 11, 121, 10201, 11111, 112211, 122221, 1002001, 1120211, 11022011, 100020001, 101212101, 122111221, 1012112101, 1100220011, 10000200001, 10111011101, 110002200011, 111221122111, 1000002000001, 1001221221001, 1012200022101, 1101202021011, 1221221221221, 10101111110101

Offset: 1

N. J. A. Sloane, Oct 22 2015

Robert Israel, Table of n, a(n) for n = 1..143
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A029984, A262607, A029985.

Intersection of A001738 and A118594.

rev3:= proc(n) local L,i; L:= convert(n,base,3); add(L[-i]*3^(i-1),i=1..nops(L)) end proc:
c3:= proc(n) local L,i; L:= convert(n,base,3); add(L[i]*10^(i-1),i=1..nops(L)) end proc:
R:= 0,1: count:= 2:
for d from 2 while count < 100 do
    if d::odd then
      V:= select(issqr, [seq(seq(a*3^((d+1)/2) + b*3^((d-1)/2)+rev3(a),b=0..2),a=3^((d-3)/2) .. 3^((d-1)/2)-1)])
    else
      V:= select(issqr, [seq(a*3^(d/2) + rev3(a), a=3^(d/2-1) .. 3^(d/2)-1)]);
    fi;
    count:= count+nops(V);
    R:= R, op(map(c3,V));
od:
R; # Robert Israel, May 19 2024

Name edited by Robert Israel, May 19 2024

A006940 Rows of Pascal's triangle mod 3.

Original entry on oeis.org

1, 11, 121, 1001, 11011, 121121, 1002001, 11022011, 121212121, 1000000001, 11000000011, 121000000121, 1001000001001, 11011000011011, 121121000121121, 1002001001002001, 11022011011022011, 121212121121212121, 1000000002000000001, 11000000022000000011, 121000000212000000121

Offset: 0

N. J. A. Sloane

nonn

Subsequence of A118594. - Chai Wah Wu, Jul 30 2025

C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 353.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A007318, A083093, A118594.

a[n_] := FromDigits[Table[Mod[Binomial[n, k], 3], {k, 0, n}]]; Array[a, 25, 0] (* Amiram Eldar, Nov 22 2018 *)

a(n)=fromdigits(apply(x->x%3, binomial(n))); \\ Michel Marcus, Nov 21 2018

from math import prod, comb
from gmpy2 import digits
def A006940(n):
    if n==0: return 1
    c, l = '', len(s:=digits(n,3))
    for k in range(m:=n+2>>1):
        t = digits(k,3).zfill(l)
        c += str(prod(comb(int(s[i]),int(t[i]))%3 for i in range(l))%3)
    return int(c+c[m-2+(n&1)::-1]) # Chai Wah Wu, Jul 30 2025

More terms from Michel Marcus, Nov 21 2018

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A118600 Palindromes in base 9 (written in base 9).

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A111575 Powers of 3 repeated four times.

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A263608 Palindromes which are base-3 representations of squares.

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A006940 Rows of Pascal's triangle mod 3.

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