A094202 -id:A094202 - cp's OEIS frontend

A095730 Primes p whose Zeckendorf-expansion A014417(p) is palindromic.

127, 197, 1949, 2137, 3323, 3821, 7253, 8117, 10243, 13183, 14947, 15131, 30941, 31721, 39607, 43691, 49207, 54773, 62213, 66413, 70141, 70429, 70607, 71089, 123457, 123923, 129023, 134039, 137699, 145391, 149381, 157219, 162523, 167759, 172357, 176237, 181253

Offset: 1

Antti Karttunen, Jun 12 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, J. Moyer: C-program for computing the initial terms of this sequence

Intersection of A000040 and A094202. Cf. A095731 for number of occurrences. A095733 shows the corresponding Fibonacci-representations.

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A349238 Reverse the digits in the Zeckendorf representation of n (A189920).

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 6, 4, 1, 9, 6, 4, 12, 1, 14, 9, 6, 19, 4, 17, 12, 1, 22, 14, 9, 30, 6, 27, 19, 4, 25, 17, 12, 33, 1, 35, 22, 14, 48, 9, 43, 30, 6, 40, 27, 19, 53, 4, 38, 25, 17, 51, 12, 46, 33, 1, 56, 35, 22, 77, 14, 69, 48, 9, 64, 43, 30, 85, 6, 61, 40, 27

Offset: 0

Kevin Ryde, Nov 11 2021

Fixed points a(n) = n are the Zeckendorf palindromes n = A094202.
Apart from a(0)=0, all terms end with a 1 digit so are "odd" A003622.
a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00 which reverses to 00..001.
A given k first occurs as a(n) = k at its reversal n = a(k), and thereafter at this n with any number of least significant 0's appended.
The equivalent reversal in binary is A030101 so that a conversion to Fibbinary (A003714) and back gives a(n) = A022290(A030101(A003714(n))).
A reverse and reverse again loses any least significant 0 digits as in A348853 so that a(a(n)) = A348853(n).

			n    = 1445 = Zeckendorf 101000101001000
a(n) =  313 = Zeckendorf 000100101000101 reversal

Kevin Ryde, Table of n, a(n) for n = 0..10000
Kevin Ryde, PARI/GP Code.

Cf. A189920 (Zeckendorf digits), A094202 (fixed points), A003622 (range), A348853 (delete trailing 0's).
Cf. A003714 (Fibbinary), A022290 (its inverse).
Cf. A343150 (reverse below MSB).
Other base reversals: A030101 (binary), A004086 (decimal).

PARI
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\\ See links.
```

def NumToFib(n): # n > 0
    f0, f1, k = 1, 1, 0
    while f0 <= n:
        f0, f1, k = f0+f1, f0, k+1
    s = ""
    while k > 0:
        f0, f1, k = f1, f0-f1, k-1
        if f0 <= n:
            s, n = s+"1", n-f0
        else:
            s = s+"0"
    return s
def RevFibToNum(s):
    f0, f1 = 1, 1
    n, k = 0, 0
    while k < len(s):
        if s[k] == "1":
            n = n+f0
        f0, f1, k = f0+f1, f0, k+1
    return n
n, a = 0, 0
print(a, end = ", ")
while n < 71:
    n += 1
    print(RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

There is a linear representation of rank 6 for this sequence. - Jeffrey Shallit, May 13 2023

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of -5 is 1111 which is palindromic.

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

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.

Original entry on oeis.org

59, 61, 69, 75, 77, 100, 105, 113, 115, 122, 128, 130, 131, 135, 136, 140, 142, 143, 148, 151, 153, 160, 162, 163, 166, 172, 177, 180, 183, 188, 191, 192, 196, 198, 200, 209, 210, 212, 215, 222, 223, 229, 230, 231, 237, 240, 249, 250, 257, 258, 263, 264, 266

Offset: 1

A.H.M. Smeets, Oct 23 2021

Zeckendorf representation version of A023108 (base 10).
For the Zeckendorf representation of numbers see A014417.
For palindromic numbers in Zeckendorf representation see A094202.
The "Reverse and Add!" operation (A349239) applied in Zeckendorf representation seems to behave similarly to the "Reverse and Add!" operation applied in any fixed-base representation. The first 53 terms are however obtained after performing 10^4 "Reverse and Add!" steps (see Python program).
For records and record-setting values in the number of "Reverse and Add!" steps see A348572 and A348571 respectively.
Do any of these numbers have a trajectory in which the Lychrel property can be proved (like 22 in base 2 as in A061561)?
Iteration steps are given by n := n+A349238(n), or n := A349239(n).
Closure of reverse operation is given by: Let Z be the regular expression for numbers in Zeckendorf representation, Z = 0|(100*)*10*, and L(Z) its corresponding regular language. Then for s in L(Z), the reversal of s is in L(0*)L(Z).
Let h be the homomorphism from Zeckendorf representation to a conventional radix representation, then addition in Zeckendorf representation, +_Z, is given by z1 +_Z z2 = h^(-1)(h(z1) + h(z2)). A direct method for addition in Zeckendorf representation is given by Ahlbach et al.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
C. Ahlbach, J. Usatine, C. Frougny, and N. Pippenger, Efficient algorithms for Zeckendorf arithmetic, Fibonacci Quart. 51, no. 3 (2013), 249-255.

Cf. A014417, A061561, A094202, A348571, A348572, A349238, A349239.

Lychrel numbers in fixed bases: A066059 (base 2), A077404 (base 3), A075420 (base 4), A023108 (base 10).

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
while n < 53:
    a += 1
    aa, sa = a, NumToFib(a)
    ar, s = RevFibToNum(sa), 0
    while aa != ar and s < 10000:
        s, aa = s+1, aa+ar
        sa = NumToFib(aa)
        ar = RevFibToNum(sa)
    if aa != ar:
        n += 1
        print(a, end = ", ")

A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3, 1, 0, 3, 2, 4, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 2, 1, 3, 3, 2, 1, 0, 2, 2, 1, 3, 1, 0, 3, 2, 4, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3

Offset: 0

Antti Karttunen, Jun 05 2004

Least number of flips of "fibits" (changing either 0 to 1 or 1 to 0 in Zeckendorf-expansion A014417(n)) so that a palindrome is produced.

			The integers 0 and 1 look as '0' and '1' also in Fibonacci-representation,
and being palindromes, a(0) and a(1) = 0.
2 has Fibonacci-representation '10', which needs a flip of other 'fibit',
that it would become a palindrome, thus a(2) = 1. Similarly 3 has representation
'100', so flipping for example the least significant fibit, we get '101',
thus a(3)=1 as well. 7 (= F(3)+F(5)) has representation '1010', which needs
two flips to produce a palindrome, thus a(7)=2. Here F(n) = A000045(n).

a(n) = A037888(A003714(n)). A094202 gives the positions of zeros. Cf. also A095732.

A329459 Numbers whose ternary and Zeckendorf representations are both palindromic.

Original entry on oeis.org

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

A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 0, 3, 7, 0, 4, 7, 0, 12, 0, 6, 10, -2, 14, 2, 8, 20, 0, 9, 15, -5, 20, 0, 9, 25, 5, 14, 20, 0, 33, 0, 14, 23, -10, 30, -3, 11, 36, 3, 17, 26, -7, 43, 10, 24, 33, 0, 40, 7, 21, 54, 0, 22, 36, -18, 46, -8, 14, 54, 0, 22, 36, -18, 62, 8, 30, 44

Offset: 0

Kevin Ryde, Nov 11 2021

Kevin Ryde, Table of n, a(n) for n = 0..10000
Kevin Ryde, PARI/GP Code

Cf. A189920 (Zeckendorf digits), A349238 (reverse), A349239 (reverse and add).
Cf. A094202 (indices of 0's).
Other bases: A055945 (binary), A056965 (decimal).

PARI
```
\\ See links.
```

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
print(a - a, end = ", ")
while n < 71:
    n += 1
    print(n - RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

a(n) = n - A349238(n).

a(n) = 2*n - A349239(n).

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]

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[#]] &]

cp's OEIS Frontend

A095730 Primes p whose Zeckendorf-expansion A014417(p) is palindromic.

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A342725 Numbers that are palindromic in base i-1.

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A349238 Reverse the digits in the Zeckendorf representation of n (A189920).

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A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

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A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.

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A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

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A329459 Numbers whose ternary and Zeckendorf representations are both palindromic.

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A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

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PARI

Python

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

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