A331191 -id:A331191 - cp's OEIS frontend

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

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[#]] &]

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);}

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

Original entry on oeis.org

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)

A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 9, 12, 15, 18, 22, 33, 44, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 145, 174, 203, 232, 261, 290, 319, 348, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 988

Offset: 1

Amiram Eldar, Feb 28 2025

The numbers of the form Fibonacci(2*k) + 1 (A055588) are all terms since A381579(A055588(0)) = 1, A381579(A055588(1)) = 2, and A381579(A055588(k)) = 10^(k-1)+1 (i.e., two 1's with k-2 0's between them) for k >= 2.

			The first 10 terms are:
   n  a(n) A381579(a(n))
   ---------------------
   1   0               0
   2   1               1
   3   2               2
   4   4              11
   5   9             101
   6  12             111
   7  15             121
   8  18             202
   9  22            1001
  10  33            1111

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

Cf. A000045, A381579.
Subsequence: A055588.
Similar sequences: A002113, A006995, A094202, A331191.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2*10^(k-1); m -= 2*f[k], s += 10^(k-1); m -= f[k]]]; PalindromeQ[s]]; Select[Range[0, 1000], q]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
isok(n) = {my(s = 0, m = n, k, d); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2*10^(k-1); m -= 2*f(k), s += 10^(k-1); m -= f(k))); d = digits(s); Vecrev(d) == d;}

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments