A014190 -id:A014190 - cp's OEIS frontend

A329459 Numbers whose ternary and Zeckendorf representations are both palindromic.

0, 1, 4, 56, 80, 203, 572, 847, 1402, 93496, 128180, 431060, 467852, 1465676, 7742920, 8727388, 8923840, 9582707, 18245944, 18304588, 25154692, 27262924, 115404434, 209060644, 763786258, 860973806, 2042328148, 4719261289, 5236838932, 18202403140, 42897493894, 77310551669

Offset: 1

Alex Ratushnyak, Nov 13 2019

Intersection of A014190 and A094202.

Chai Wah Wu, Table of n, a(n) for n = 1..56 (n = 1..43 from Daniel Suteu)

Cf. A014190, A094202, A060792, A095309.

a(24)-a(32) from Daniel Suteu, Nov 16 2019

A338828 For any number with ternary representation (t(1), t(2), ..., t(k)), the ternary representation of a(n) is (abs(t(1)-t(k)), abs(t(2)-t(k-1)), ..., abs(t(k)-t(1))).

Original entry on oeis.org

0, 0, 0, 4, 0, 4, 8, 4, 0, 10, 0, 10, 10, 0, 10, 10, 0, 10, 20, 10, 0, 20, 10, 0, 20, 10, 0, 28, 0, 28, 40, 12, 40, 52, 24, 52, 40, 12, 40, 28, 0, 28, 40, 12, 40, 52, 24, 52, 40, 12, 40, 28, 0, 28, 56, 28, 0, 68, 40, 12, 80, 52, 24, 68, 40, 12, 56, 28, 0, 68

Offset: 0

Rémy Sigrist, Nov 11 2020

Leading zeros are ignored.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A014190, A175919 (binary analog), A338827 (decimal analog).

a:= n-> (l-> (h-> add(h[j]*3^(j-1), j=1..nops(h)))([seq(
    abs(l[i]-l[-i]), i=1..nops(l))]))(convert(n, base, 3)):
seq(a(n), n=0..70);  # Alois P. Heinz, Nov 12 2020

a(n, base=3) = my (d=digits(n, base)); fromdigits(abs(d-Vecrev(d)), base)

a(n) = 0 iff n is a palindrome in base 3 (A014190).

A352507 Number whose representation in the base of Catalan numbers (A014418) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 15, 22, 43, 48, 53, 59, 64, 69, 133, 152, 171, 177, 196, 215, 430, 444, 458, 477, 491, 505, 524, 538, 552, 564, 578, 592, 611, 625, 639, 658, 672, 686, 1431, 1487, 1543, 1568, 1624, 1680, 1705, 1761, 1817, 1862, 1918, 1974, 1999, 2055, 2111, 2136

Offset: 1

Amiram Eldar, Mar 19 2022

The partial sums of the Catalan numbers with positive index (A014138) are terms, since the representation of A014138(n) is n 1's.

			The first 10 terms are:
   n  a(n)  A014418(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3             11
   4     6            101
   5     8            111
   6    15           1001
   7    22           1111
   8    43          10001
   9    48          10101
  10    53          10201

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

Cf. A000108, A014418.
Subsequences: A014138, A141351 \ {2}.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; PalindromeQ @ IntegerDigits[Total[4^(s - 1)], 4]]; Select[Range[0, 2000], q]

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

A354886 Numbers that are palindromes in both ternary and balanced ternary representations with representations that are different.

Original entry on oeis.org

16, 52, 160, 484, 1312, 1456, 3904, 4372, 11680, 12688, 13120, 35008, 37960, 39364, 104992, 106288, 113776, 116800, 118096, 314944, 319156, 341224, 350080, 354292, 944800, 948688, 957760, 1023568, 1027456, 1049920, 1058992, 1062880, 2834368, 2847004, 2873572

Offset: 1

Amiram Eldar, Jun 10 2022

Is a(n) == 16 (mod 18) for all n?

			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

Equals A354885 \ A005836.

Cf. A007089, A014190, A091077, A134027, A140267.

Select[Range[0, 3*10^6], PalindromeQ[d3 = IntegerDigits[#, 3]] && PalindromeQ[db3 = balTernDigits[#]] && d3 != db3 &] (* using balTernDigits by Robert G. Wilson v at A134027 *)

a(n) = A091077(n) / 4. - Hugo Pfoertner, Jun 10 2022

A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.

Original entry on oeis.org

0, 1, 2, 5, 7, 10, 13, 15, 23, 28, 34, 36, 52, 57, 65, 75, 81, 89, 91, 117, 128, 146, 159, 175, 185, 198, 204, 217, 233, 235, 277, 295, 327, 369, 379, 400, 426, 442, 463, 473, 494, 520, 526, 547, 573, 589, 610, 612, 680, 709, 761, 829, 848, 916, 945, 989, 1023

Offset: 1

Amiram Eldar, Jul 01 2023

Includes all the odd-indexed Fibonacci numbers (A001519), since the Wythoff representation of Fibonacci(1) is 1 and the Wythoff representation of Fibonacci(2*n+1), for n >= 1, is n 0's.
A157725(n) = Fibonacci(n) + 2 is a term for n >= 4, since its Wythoff representation is n-4 1's between 2 0's.
A232970 is a subsequence since the Wythoff representation of A232970(n) = (Fibonacci(3*n+1) + 1)/2 is n 0's and n-1 1's interleaved.

			The first 10 terms are:
   n  a(n)  A317208(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     5             22
   5     7            212
   6    10           2112
   7    13            222
   8    15          21112
   9    23         211112
  10    28          21212

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

Cf. A001519, A157725, A189921, A232970, A317208.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A352507.

z[n_] := Floor[(n + 1)*GoldenRatio] - n - 1; h[n_] := z[n] - z[n - 1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; w[0] = {0}; Select[Range[0, 1000], PalindromeQ[w[#]] &]

A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 15, 18, 21, 22, 30, 34, 42, 44, 49, 58, 63, 66, 71, 80, 85, 86, 102, 110, 126, 130, 146, 154, 170, 172, 183, 198, 209, 218, 229, 244, 255, 258, 269, 284, 295, 304, 315, 330, 341, 342, 374, 390, 422, 430, 462, 478, 510, 514, 546, 562, 594

Offset: 1

Amiram Eldar, Jul 14 2023

The even-indexed Jacobsthal numbers A001045(2*n) = A002450(n) = (4^n-1)/3, for n >= 1, are terms since their representation is 2*n-1 1's.
A001045(2*n+1) - 1 = A020988(n) = (2/3)*(4^n-1) is a term for n >= 1, since its representation is 2*n 1's.
A001045(n) + 1 = A128209(n) is a term for n >= 0, since its representation for n = 0 is 1 and its representation for n >= 1 is n-1 0's between 2 1's.
A160156(n) is a term for n >= 0 since its representation is n 0's interleaved with n+1 1's.

			The first 10 terms are:
   n  a(n)  A280049(a(n))
  --  ----  -------------
   1     1              1
   2     2             11
   3     4            101
   4     5            111
   5     6           1001
   6    10           1111
   7    12          10001
   8    15          10101
   9    18          11011
  10    21          11111

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

Cf. A001045, A002450, A003159, A020988, A280049.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Position[Select[Range[1000], EvenQ[IntegerExponent[#, 2]] &], _?(PalindromeQ[IntegerDigits[#, 2]] &)] // Flatten

s(n) = if(n < 2, n > 0, n = s(n-1); until(valuation(n, 2)%2 == 0, n++); n); \\ A003159
is(n) = {my(d = binary(s(n))); d == Vecrev(d);}

A043262 Sum of digits of n-th base 3 palindrome.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 3, 4, 4, 5, 6, 2, 4, 6, 4, 6, 8, 2, 3, 4, 4, 5, 6, 6, 7, 8, 4, 5, 6, 6, 7, 8, 8, 9, 10, 2, 4, 6, 4, 6, 8, 6, 8, 10, 4, 6, 8, 6, 8, 10, 8, 10, 12, 2, 3, 4, 4, 5, 6, 6, 7, 8, 4, 5, 6, 6, 7, 8, 8, 9, 10, 6, 7, 8, 8, 9, 10, 10, 11, 12, 4, 5, 6, 6, 7, 8

Offset: 1

Clark Kimberling

Cf. A014190 (base 3 palindromes)

Total/@Select[IntegerDigits[Range[0,2000],3],#==Reverse[#]&] (* Harvey P. Dale, Aug 08 2019 *)

from gmpy2 import digits
def A043262(n):
    if n == 1: return 0
    y = 3*(x:=3**(len(digits(n>>1, 3))-1))
    return int((s:=digits(n-x,3))[-1])+(sum(int(d) for d in s[:-1])<<1) if nChai Wah Wu, Jul 24 2024

A260184 Numbers n written in base 10 that are palindromic in exactly three bases b, 2 <= b <= 10 and not simultaneously bases 2, 4 and 8.

Original entry on oeis.org

9, 10, 21, 40, 55, 80, 85, 100, 130, 154, 164, 178, 191, 203, 235, 242, 255, 257, 273, 282, 292, 300, 328, 400, 455, 585, 656, 819, 910, 2709, 6643, 8200, 14762, 32152, 53235, 74647, 428585, 532900, 1181729, 1405397, 4210945, 5259525, 27711772, 719848917, 43253138565

Offset: 1

Giovanni Resta and Robert G. Wilson v, Jul 17 2015

			273 is in the sequence because 100010001_2 = 101010_3 = 10101_4 = 2043_5 = 1133_6 = 540_7 = 421_8 = 333_9 = 273_10 and three of the bases, namely 2, 4 & 9, yield palindromes.

Cf. A214426, A214425.

(* see A214425 and set all terms as lst, then *)
gQ[n_] := Count[ palQ[n,#] & /@ {2, 4, 8}, True];
Select[ lst, gQ[#] != 3 &]

The intersection of A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955 & A002113 which yields just three members, not simultaneously bases 2, 4 and 8.

A309574 n-th prime minus its ternary (base 3) reversal.

Original entry on oeis.org

0, 2, -2, 2, -8, 0, -8, 8, 0, -26, -6, 6, -26, -6, -14, -26, -6, 14, 26, -6, 38, 26, -80, -128, -48, -80, -24, -128, 24, -80, 24, -80, -32, 24, -56, 0, 24, 80, -24, 0, -48, 80, 24, 80, -24, 104, 80, 80, 48, 104, 0, 24, 80, -398, -338, -278, -434, 18, -138

Offset: 1

Samuel M. A. Luque Astorga, Aug 08 2019

a(n) = 0 if and only if n is palindromic in base 3 - if and only if A000040(n) is in A014190.
As it occurs in its binary cousin, we observe that a scatter plot of this sequence shows parallelograms.
All terms are even. - Alois P. Heinz, Aug 08 2019

Robert Israel, Table of n, a(n) for n = 1..10000
Samuel M. A. Luque, C++ Program (.cpp file) to generate this sequence in whichever m-th base you choose.
Samuel M. A. Luque, Scatter plot of a(n) up to n = 8132 (primes up to 50 000).

Cf. A000040, A014190, A030102, A055947, A265326.

a:= n-> (p-> p-(l->add(l[-i]*3^(i-1), i=1..nops(l))
        )(convert(p, base, 3)))(ithprime(n)):
seq(a(n), n=1..61);  # Alois P. Heinz, Aug 08 2019

(# - IntegerReverse[#,3]) &@ Prime@ Range@ 60 (* Giovanni Resta, Aug 09 2019 *)

a(n) = my(p=prime(n)); p - fromdigits(Vecrev(digits(p, 3)), 3); \\ Michel Marcus, Aug 09 2019

from sympy import primerange
def rev(n, b):
    m = 0
    while n > 0:
        m, n = m*b+n%b, n//b
    return m
n, aa = 1, 1
while n <20:
    if aa in primerange(1,200):
        print(n, aa-rev(aa, 3))
        n = n+1
    aa = aa+1 # A.H.M. Smeets, Aug 09 2019

a(n) = A000040(n) - A030102(A000040(n)).

a(n) = A055947(A000040(n)).

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A329459 Numbers whose ternary and Zeckendorf representations are both palindromic.

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A338828 For any number with ternary representation (t(1), t(2), ..., t(k)), the ternary representation of a(n) is (abs(t(1)-t(k)), abs(t(2)-t(k-1)), ..., abs(t(k)-t(1))).

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A352507 Number whose representation in the base of Catalan numbers (A014418) is palindromic.

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

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A354886 Numbers that are palindromes in both ternary and balanced ternary representations with representations that are different.

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A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.

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A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

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A043262 Sum of digits of n-th base 3 palindrome.

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