A134027 -id:A134027 - cp's OEIS frontend

A014190 Palindromes in base 3 (written in base 10).

0, 1, 2, 4, 8, 10, 13, 16, 20, 23, 26, 28, 40, 52, 56, 68, 80, 82, 91, 100, 112, 121, 130, 142, 151, 160, 164, 173, 182, 194, 203, 212, 224, 233, 242, 244, 280, 316, 328, 364, 400, 412, 448, 484, 488, 524, 560, 572, 608, 644, 656, 692, 728, 730, 757

Offset: 1

N. J. A. Sloane

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

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.
Eric Weisstein's World of Mathematics, Palindromic Number.
Eric Weisstein's World of Mathematics, Ternary.
Index entries for sequences that are an additive basis, order 3.

Cf. A007089, A118594, A134027, A330312 (first differences).

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

[n: n in [0..800] | Intseq(n, 3) eq Reverse(Intseq(n, 3))]; // Vincenzo Librandi, Sep 09 2015

isA014190 := proc(n)
    local L;
    L := convert(n,base,3) ;
    ListTools[Reverse](L) = L ;
end proc:
for n from 0 to 500 do
    if isA014190(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2015

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,3], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

ispal(n,b=3)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def A014190(n):
    if n == 1: return 0
    y = 3*(x:=3**(len(digits(n>>1,3))-1))
    return int((c:=n-x)*x+int(digits(c,3)[-2::-1]or'0',3) if nChai Wah Wu, Jun 13 2024

[n for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

Sum_{n>=2} 1/a(n) = 2.61676111... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A134021 Length of n in balanced ternary representation.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Oct 19 2007

Shifted variant of A064099.

			100 = 1*3^4+1*3^3-1*3^2+0*3^1+1*3^0: a(100) = |++-0+| = 5.
200 = 1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0: a(200) = |+-+++-| = 6.
300 = 1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0: a(300) = |++-0+0| = 6.

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced Ternary.

Cf. A005812, A059095, A081604, A134022, A134023, A134024, A134025, A134026, A134027, A134028, A134421.

a[n_] := Ceiling[Log[3, 2*n+1]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Apr 03 2025 *)

def a(n):
    if n==0: return 1
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

For n > 0: a(n) = ceiling(log(2*n+1)/log(3)).
a(n) = A134022(n) + A134023(n) + A134024(n).
0 <= a(n) - A081604(n) <= 1.
a(A134025(n)) = A081604(A134025(n)); a(A134026(n)) = A081604(A134026(n))+1.
a(A134027(n)) = a(n); a(abs(A134028(n))) <= a(n).
a(n) = A064099(n-1) for n>1.
n = Sum_{k=0..a(n)-1} (A059095(A134421(n)-2-k)*3^k), for n > 0. - Reinhard Zumkeller, Oct 25 2007
a(n) = A005812(n) + A134023(n).

A134028 Reversal of n in balanced ternary representation.

Original entry on oeis.org

0, 1, -2, 1, 4, -11, -2, 7, -8, 1, 10, -5, 4, 13, -38, -11, 16, -29, -2, 25, -20, 7, 34, -35, -8, 19, -26, 1, 28, -17, 10, 37, -32, -5, 22, -23, 4, 31, -14, 13, 40, -119, -38, 43, -92, -11, 70, -65, 16, 97, -110, -29, 52, -83, -2, 79, -56, 25, 106, -101, -20, 61, -74, 7, 88, -47, 34, 115, -116, -35, 46, -89, -8, 73, -62, 19, 100

Offset: 0

Reinhard Zumkeller, Oct 19 2007

As the graph demonstrates, the sequence makes large negative steps at terms (3^i+1)/2. These steps divide the graph into conspicuous blocks. - N. J. A. Sloane, Jul 03 2016

			20 = 1*3^3 - 1*3^2 + 1*3^1 - 1*3^0 == '+-+-'
=> a(20) = -1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = -20;
21 = 1*3^3 - 1*3^2 + 1*3^1 + 0*3^0 == '+-+0'
=> a(21) = 0*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 7;
22 = 1*3^3 - 1*3^2 + 1*3^1 + 1*3^0 == '+-++'
=> a(22) = 1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 34;
23 = 1*3^3 + 0*3^2 - 1*3^1 - 1*3^0 == '+0--'
=> a(23) = -1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -35;
24 = 1*3^3 + 0*3^2 - 1*3^1 + 0*3^0 == '+0-0'
=> a(24) = 0*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -8;
25 = 1*3^3 + 0*3^2 - 1*3^1 + 1*3^0 == '+0-+'
=> a(25) = 1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = 19.

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Reversal
Wikipedia, Balanced Ternary

Cf. A117966 (balanced ternary representation), A030102, A134021, A274107.

A134027 gives the numbers whose balanced ternary representation is palindromic.

def a(n):
    if n==0: return 0
    s=[]
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s.append(x)
    l=s[::-1]
    return sum(l[i]*3**i for i in range(len(l)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 10 2017

a(A134027(n)) = A134027(n);

A134021(ABS(a(n))) <= A134021(n).

A233010 In balanced ternary notation, either a palindrome or becomes a palindrome if trailing 0's are omitted.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 10, 12, 13, 16, 21, 27, 28, 30, 36, 39, 40, 43, 48, 52, 61, 63, 73, 81, 82, 84, 90, 91, 103, 108, 112, 117, 120, 121, 124, 129, 144, 156, 160, 183, 189, 196, 208, 219, 243, 244, 246, 252, 270, 273, 280, 292, 309, 324, 328, 336, 351, 360, 363

Offset: 1

Lei Zhou, Dec 13 2013

Symmetric strings of -1, 0, and 1, including as many leading as trailing zeros.

			10 is included since in balanced ternary notation 10 = (101)_bt is a palindrome;
144 is included since 144 = (1TT100)_bt, where we use T to represent -1.  When trailing zeros removed, 1TT1 is a palindrome.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A002113, A061917, A006995, A057890, A134027

BTDigits[m_Integer, g_] :=
Module[{n = m, d, sign, t = g},
  If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
   d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
   While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign;
   t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
BTpaleQ[n_Integer] := Module[{t, trim = n/3^IntegerExponent[n, 3]},
  t = BTDigits[trim, {0}]; t == Reverse[t]];
Select[Range[0, 363], BTpaleQ[#] &]

A233571 In balanced ternary notation, reverse digits of a(n) equals to -a(n).

Original entry on oeis.org

0, 2, 8, 20, 26, 32, 56, 80, 104, 146, 164, 182, 224, 242, 260, 302, 320, 338, 416, 488, 560, 656, 728, 800, 896, 968, 1040, 1172, 1226, 1280, 1406, 1460, 1514, 1640, 1694, 1748, 1898, 1952, 2006, 2132, 2186, 2240, 2366, 2420, 2474, 2624, 2678, 2732, 2858

Offset: 1

Lei Zhou, Dec 13 2013

			In balanced ternary notation, 8=(10T)_bt, where we use T to represent -1.  Reverse digits of (10T)_bt is (T01)_bt = -8. So 8 is in this sequence.
Similarly, 2240 = (1001T00T)_bt, whose reverse digits is (T00T1001)_bt = -2240.  So 2240 is in this sequence.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A002113, A061917, A006995, A057890, A134027, A233010

BTDigits[m_Integer, g_] :=
Module[{n = m, d, sign, t = g},
  If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
   d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
   While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign;
   t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
ct = 1; n = 0; m = 0; dg = 0; switch = 1; res = {0}; While[ct < 50, n++;
  bits = BTDigits[n, {0}]; If[lb = Length[bits]; lb > dg,
  If[switch == 0, n = m; switch = 1; OK = 0, dg = lb; m = n - 1;
   switch = 0; OK = 1], OK = 1];  If[OK == 1, rbt = -Reverse[bits];
  If[switch == 1, nb = Join[bits, {0}], nb = bits];
  nb = Join[nb, rbt]; nb = Reverse[nb]; data = 0;
  Do[data = data + 3^(i - 1)*nb[[i]], {i, 1, Length[nb]}]; ct++;
  AppendTo[res, data]]]; res

A233572 In balanced ternary notation, if prepending same numbers of zeros, reverse digits of a(n) equals to -a(n).

Original entry on oeis.org

0, 2, 6, 8, 18, 20, 24, 26, 32, 54, 56, 60, 72, 78, 80, 96, 104, 146, 162, 164, 168, 180, 182, 216, 224, 234, 240, 242, 260, 288, 302, 312, 320, 338, 416, 438, 486, 488, 492, 504, 540, 546, 560, 648, 656, 672, 702, 720, 726, 728, 780, 800, 864, 896, 906, 936

Offset: 1

Lei Zhou, Dec 13 2013

A233571 is a subset of this sequence.

			In balanced ternary notation, 18 = (1T00)_bt, where we use T to represent -1.  Patching two zeros before it, (1T00)_bt=(001T00)_bt.  The reverse digits of (001T00)_bt is (00T100)_bt = -18.  So 18 is in this sequence.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A002113, A061917, A006995, A057890, A134027, A233010, A233571

BTDigits[m_Integer, g_] :=
Module[{n = m, d, sign, t = g},
  If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
   d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
   While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign;
   t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
BTrteQ[n_Integer] :=
Module[{t, trim = n/3^IntegerExponent[n, 3]},
  t = BTDigits[trim, {0}]; DeleteDuplicates[t + Reverse[t]] == {0}];
sb = Select[Range[0, 950], BTrteQ[#] &]

A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 11, 6, 9, 12, 7, 10, 13, 14, 23, 32, 17, 26, 35, 20, 29, 38, 15, 24, 33, 18, 27, 36, 21, 30, 39, 16, 25, 34, 19, 28, 37, 22, 31, 40, 41, 68, 95, 50, 77, 104, 59, 86, 113, 44, 71, 98, 53, 80, 107, 62, 89, 116, 47, 74, 101, 56, 83, 110, 65, 92

Offset: 0

Kevin Ryde, Feb 19 2022

Self-inverse permutation with swaps confined to terms of a given digit length (A134021) so within blocks n = (3^k+1)/2 .. (3^(k+1)-1)/2.
Can extend to negative n by a(-n) = -a(n).
A072998 is balanced ternary coded in decimal digits so that reversal except first digit of A072998(n) is at A072998(a(n)).  Similarly its ternary equivalent A157671, and also A132141 ternary starting with 1.
These sequences all have a fixed initial digit followed by all ternary strings which is the reversed part.  A007932 is such strings as decimal digits 1,2,3 but it omits the empty string so the whole reversal of A007932(n) is at A007932(a(n+1)-1).
Fixed points a(n) = n are where n in balanced ternary is a palindrome apart from its initial 1.  These are the full balanced ternary palindromes with their least significant 1 removed, so all n = (A134027(m)-1)/3 for m>=2.

			n    = 224 = balanced ternary 1,  0, -1, 1,  0, -1
                      reverse     ^^^^^^^^^^^^^^^^
a(n) = 168 = balanced ternary 1, -1,  0, 1, -1,  0

Cf. A059095 (balanced ternary), A134028 (full reverse), A134027 (palindromes).
Cf. A072998, A157671, A132141, A007932.
In other bases: A059893 (binary), A343150 (Zeckendorf), A343152 (lazy Fibonacci).

a(n) = if(n==0,0, my(k=if(n,logint(n<<1,3)), s=(3^k+1)>>1); s + fromdigits(Vec(Vecrev(digits(n-s,3)),k),3));

A224502 Prime numbers (together with one) whose representation in balanced ternary are palindromes.

Original entry on oeis.org

1, 7, 13, 43, 61, 73, 103, 367, 421, 457, 547, 601, 613, 757, 859, 1039, 1093, 3823, 4021, 4561, 4723, 4759, 5743, 6211, 6373, 6481, 6949, 7219, 7489, 7933, 8563, 8941, 9103, 9679, 29527, 30013, 31147, 31741, 33037, 35251, 36061, 36097, 36583, 37717, 39607, 41011, 42667, 43963, 44773, 45691, 47581, 49201

Offset: 1

Malachi de Ælfweald, Apr 08 2013

Intersection of A006005 and A134027.

			For n=5, a(5)=61 and in balanced ternary notation is 1ī1ī1.

Malachi de Ælfweald and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 199 terms from Malachi de Ælfweald)
Wikipedia, Balanced ternary

bt(k,n)={
    sum(i=0,(n-1)\2,
        my(t=k%3-1);
        k\=3;
        n--;
        if(n==i,3^n,3^i+3^n)*t
    )
};
do(N)={
    my(v=List([1]),t);
    for(n=1,N,
        forstep(k=2,3^((n+1)\2)-1,3,
            t=bt(k,n);
            if(isprime(t),listput(v,t))
        )
    );
    vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Apr 08 2013

A233573 Number of ways n can be partitioned as A233010(i)+A233572(j), where i,j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 4, 2, 2, 3, 2, 1, 3, 2, 1, 4, 2, 2, 4, 2, 2, 3, 2, 1, 6, 3, 4, 5, 2, 3, 5, 3, 3, 7, 1, 3, 4, 2, 3, 4, 1, 2, 5, 2, 3, 5, 1, 2, 2, 2, 2, 6, 1, 4, 3, 3, 3, 6, 4, 2, 7, 3, 3, 4, 4, 3, 5, 3, 2, 6, 3, 1, 4, 3, 1, 2, 4, 2, 9, 4, 4, 7, 3, 2

Offset: 0

Lei Zhou, Dec 13 2013

Number in sequence A233010 takes palindrome form when written in balanced ternary notation and left patch same number of zeros as number of trailing zeros, which means that if we slice the balanced ternary notation number with left zero padding in the middle, the left part is a mirrored image of the right part. This is a feature like an even function.
Number in sequence A233571 takes "reversed palindrome form" when written in balanced ternary notation and left patch same number of zeros as number of trailing zeros, which means that if we slice the balanced ternary notation number with left zero padding in the middle, the left part is a mirrored image of the right part with sign changes. This is a feature like an odd function.
This sequence gives the number of possibilities of such partitions.
In the first 10000 terms, 152 zeros found.

			0=0+0=A233010(1)+A233572(1). This is the only valid partition by definition. So a(0)=1.
3=3+0=A233010(3)+A233572(1), as well 3=1+2=A233010(2)+A233572(2).
  Two valid partitions found. So a(3)=2.
9=9+0=7+2=3+6=1+8, four valid partitions found. So a(9)=4.

Lei Zhou, Table of n, a(n) for n = 0..10000

Cf. A002113, A061917, A006995, A057890, A134027, A233010, A233571, A233572

BTDigits[m_Integer, g_] :=
(*This is to determine digits of a number in balanced ternary notation.*)
Module[{n = m, d, sign, t = g},
  If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
   d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
   While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign;
     t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
BTpaleQ[n_Integer] :=
(*This is to query if a number is an element of sequence A233010.*)
Module[{t, trim = n/3^IntegerExponent[n, 3]},
  t = BTDigits[trim, {0}]; t == Reverse[t]];
BTrteQ[n_Integer] :=
(*This is to query if a number is an element of sequence A233572.*)
Module[{t, trim = n/3^IntegerExponent[n, 3]},
  t = BTDigits[trim, {0}]; DeleteDuplicates[t + Reverse[t]] == {0}];
sa = Select[Range[0, 11000], BTpaleQ[#] &];
(*This is to generate a limited list of A233010.*)
sb = Select[Range[0, 11000], BTrteQ[#] &];
(*This is to generate a limited list of A233572.*)
range = 86; Table[ct = 0; i1 = 0;
While[i1++; sa[[i1]] <= n, i2 = 0;
  While[i2++; (sa[[i1]] + sb[[i2]]) <= n,
   If[(sa[[i1]] + sb[[i2]]) == n, ct++]]]; ct, {n, 0, range}]

A354885 Numbers that are palindromes in both ternary and balanced ternary representations.

Original entry on oeis.org

0, 1, 4, 10, 13, 16, 28, 40, 52, 82, 91, 112, 121, 160, 244, 280, 328, 364, 484, 730, 757, 820, 847, 976, 1003, 1066, 1093, 1312, 1456, 2188, 2296, 2440, 2548, 2920, 3028, 3172, 3280, 3904, 4372, 6562, 6643, 6832, 6913, 7300, 7381, 7570, 7651, 8752, 8833, 9022

Offset: 1

Amiram Eldar, Jun 10 2022

			4 is a term since its ternary and balanced ternary representation are both 11 which is a palindrome.
16 is a term since its ternary representation is 121 and its balanced ternary representation (with 2 standing for the -1 digit) is 1221, and both are palindromes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A014190 and A134027.
Subsequences: A003462, A034472 \ {2}, A354886.
Cf. A007089, A140267.

Select[Range[0, 10^4], PalindromeQ[IntegerDigits[#, 3]] && PalindromeQ @ balTernDigits[#] &] (* using balTernDigits by Robert G. Wilson v at A134027 *)

is(n) = {my (t=digits(n,3)); if (t==Vecrev(t), my (b=[], d); while (n, b=concat(d=[0,1,-1][1+n%3], b); n=(n-d)/3); b==Vecrev(b), 0)} \\ Rémy Sigrist, Jun 10 2022

cp's OEIS Frontend

A014190 Palindromes in base 3 (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A134021 Length of n in balanced ternary representation.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A134028 Reversal of n in balanced ternary representation.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Python

Formula

A233010 In balanced ternary notation, either a palindrome or becomes a palindrome if trailing 0's are omitted.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A233571 In balanced ternary notation, reverse digits of a(n) equals to -a(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A233572 In balanced ternary notation, if prepending same numbers of zeros, reverse digits of a(n) equals to -a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI