A014190 -id:A014190 - cp's OEIS frontend

A354884 Numbers whose skew binary representation (A169683) is palindromic.

0, 1, 2, 4, 8, 11, 16, 26, 32, 39, 50, 57, 64, 86, 98, 120, 128, 143, 166, 181, 194, 209, 232, 247, 256, 302, 326, 372, 386, 432, 456, 502, 512, 543, 590, 621, 646, 677, 724, 755, 770, 801, 848, 879, 904, 935, 982, 1013, 1024, 1118, 1166, 1260, 1286, 1380, 1428

Offset: 1

Amiram Eldar, Jun 10 2022

The sequence of powers of 2 (A000079) is a subsequence since A169683(1) = 1, A169683(2) = 2, and for n > 2 A169683(2^n) = 10..01 with n-1 0's between the two 1's.
A000295 is a subsequence since A169683(A000295(0)) = A169683(A000295(1)) = 0 and for n>1 A169683(A000295(n)) is a repunit with n-1 1's.
A144414 is a subsequence since A169683(A144414(1)) = 1 and for n>1 A169683(A144414(n)) = 1010..01 with n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A169683(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    2               2
   4    4              11
   5    8             101
   6   11             111
   7   16            1001
   8   26            1111
   9   32           10001
  10   39           10101

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

Cf. A169683.
Subsequences: A000079, A000295, A144414.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

f[0] = 0; f[n_] := Module[{m = Floor@Log2[n + 1], d = n, pos}, Reap[While[m > 0, pos = 2^m - 1; Sow@Floor[d/pos]; d = Mod[d, pos]; --m;]][[2, 1]] // FromDigits]; Select[Range[0, 15000], PalindromeQ[f[#]] &] (* after N. J. A. Sloane at A169683 *)

A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 13, 15, 18, 21, 23, 34, 36, 40, 45, 50, 55, 66, 71, 89, 91, 95, 108, 113, 120, 128, 136, 144, 159, 176, 196, 204, 233, 235, 239, 261, 273, 286, 291, 298, 319, 327, 338, 351, 364, 377, 400, 426, 464, 490, 518, 550, 563, 610, 612, 616, 654, 667

Offset: 1

Amiram Eldar, Jul 07 2023

The positive Fibonacci numbers (A000045) are terms since the Stolarsky representation of Fibonacci(1) = Fibonacci(2) is 0 and the Stolarsky representation of Fibonacci(n) is n-2 1's for n >= 3.

Fiboancci(2*n+1) + 2 is a term for n >= 3, since its Stolarsky representation is n-1 0's between two 1's.

			The first 10 terms are:
   n  a(n)  A364121(a(n))
  --  ----  -------------
   1     1  0
   2     2  1
   3     3  11
   4     5  111
   5     6  101
   6     8  1111
   7    13  11111
   8    15  1001
   9    18  11011
  10    21  111111

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

Cf. A000045, A200648, A200649, A200650, A200651, A200714, A364121.

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

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolPalQ[n_]:= PalindromeQ[stol[n]]; Select[Range[700], stolPalQ]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
is(n) = {my(s = stol(n)); s == Vecrev(s);}

A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 20, 22, 27, 36, 41, 44, 60, 68, 84, 86, 97, 112, 123, 132, 143, 158, 169, 172, 204, 220, 252, 260, 292, 308, 340, 342, 363, 396, 417, 432, 453, 486, 507, 516, 537, 570, 591, 606, 627, 660, 681, 684, 748, 780, 844, 860, 924, 956, 1020, 1028

Offset: 1

Amiram Eldar, Jul 21 2023

A128209(n) = A001045(n) + 1 is a term for n >= 3, since its representation is two 1's with n-3 0's between them.
A084639(n) is a term for n >= 1 since its representation is n 1's.
A014825(n) is a term for n >= 1 since its representation is n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A265747(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     4             11
   5     6            101
   6     9            111
   7    12           1001
   8    20           1111
   9    22          10001
  10    27          10101

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

Cf. A001045, A014825, A084639, A128209, A265747.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A364214.

palJacobQ[n_] := PalindromeQ[A265747[n]]; Select[Range[0, 1000], palJacobQ] (* using A265747[n] *)

is(n) = {my(dig = digits(A265747(n))); dig == Vecrev(dig);} \\ using A265747(n)

A319595 Numbers in base 10 that are palindromic in bases 3, 7, and 9.

Original entry on oeis.org

0, 1, 2, 4, 8, 40, 100, 164, 328, 400, 8200, 14762, 532900

Offset: 1

Jeremias M. Gomes, Sep 23 2018

Intersection of A014190, A029954, and A029955.

No other terms < 10^17. It is likely that there are no more terms. - Chai Wah Wu, Mar 20 2020

			400 = 112211_3 = 1111_7 = 484_9.

Cf. A014190 (base 3), A029954 (base 7), and A029955 (base 9).

[n for n in (0..100000) if Word(n.digits(3)).is_palindrome() and Word(n.digits(7)).is_palindrome() and Word(n.digits(9)).is_palindrome()]

A331192 Numbers whose Zeckendorf representation (A014417) and dual Zeckendorf representation (A104326) are both palindromic.

Original entry on oeis.org

0, 1, 4, 6, 12, 22, 33, 64, 88, 174, 232, 462, 609, 1216, 1596, 3190, 4180, 8358, 10945, 21888, 28656, 57310, 75024, 150046, 196417, 392832, 514228, 1028454, 1346268, 2692534, 3524577, 7049152, 9227464, 18454926, 24157816, 48315630, 63245985, 126491968, 165580140

Offset: 1

Amiram Eldar, Jan 11 2020

Apparently union of numbers of the form F(2*k - 1) - 1 (k > 0) and numbers of the form 2 * F(2*k - 1) - 4 (k > 1), where F(m) is the m-th Fibonacci number.

The numbers of the form F(2*k - 1) - 1 have the same Zeckendorf and dual Zeckendorf representations. For k > 1 the representation is 1010...01, k-1 1's interleaved with k-2 0's.

			6 is a term since its Zeckendorf representation, 1001, and its dual Zeckendorf representation, 111, are both palindromic.

Intersection of A094202 and A331191.

Cf. A000045, A000071, A002113, A006995, A014190, A027941, A048268, A060792, A095309, A104326, A329459.

mirror[dig_, s_] := Join[dig, s, Reverse[dig]];
select[v_, mid_] := Select[v, Length[#] == 0 || Last[#] != mid &];
fib[dig_] := Plus @@ (dig*Fibonacci[Range[2, Length[dig] + 1]]);
ndig = 12; pals1 = Rest[IntegerDigits /@ FromDigits /@ Select[Tuples[{0, 1}, ndig], SequenceCount[#, {1, 1}] == 0 &]];
zeckPals = Union @ Join[{0, 1}, fib /@ Join[mirror[#, {}] & /@ (select[pals1, 1]), mirror[#, {1}] & /@ (select[pals1, 1]), mirror[#, {0}] & /@ pals1]];
pals2 = Join[{{}}, Rest[Select[IntegerDigits[Range[0, 2^ndig - 1], 2], SequenceCount[#, {0, 0}] == 0 &]]];
dualZeckPals = Union@Join[{0}, fib /@ Join[mirror[#, {}] & /@ (select[pals2, 0]), mirror[#, {0}] & /@ (select[pals2, 0]), mirror[#, {1}] & /@ pals2]];
Intersection[zeckPals, dualZeckPals]

cp's OEIS Frontend

A354884 Numbers whose skew binary representation (A169683) is palindromic.

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A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

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PARI

A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

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PARI

A319595 Numbers in base 10 that are palindromic in bases 3, 7, and 9.

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Sage

A331192 Numbers whose Zeckendorf representation (A014417) and dual Zeckendorf representation (A104326) are both palindromic.

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