A094202 -id:A094202 - cp's OEIS frontend

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

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);}

A286710 Numbers n whose Zeckendorf representation is of the form ww, for w a nonempty block of digits.

Original entry on oeis.org

7, 16, 39, 54, 97, 120, 134, 246, 282, 304, 340, 376, 631, 688, 723, 780, 837, 872, 929, 964, 1631, 1722, 1778, 1869, 1960, 2016, 2107, 2163, 2254, 2345, 2401, 2492, 2583, 4236, 4382, 4472, 4618, 4764, 4854, 5000, 5090, 5236, 5382, 5472, 5618, 5764, 5854, 6000, 6090, 6236, 6382, 6472, 6618, 6708, 11035, 11270, 11415

Offset: 1

Jeffrey Shallit, May 13 2017

The Zeckendorf representation of an integer n expresses n as a sum of non-adjacent Fibonacci numbers. It can be expressed as a word over {0,1} giving the coefficients, starting with the most significant digit.

			The representation of 7 is 1010, which is of the form ww with w = 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A014417, A094202 (the same sequence, but for palindromes).

F:= [seq(combinat:-fibonacci(i),i=2..21)]:
ext:= proc(L)
  if L[2] = 0 then [0,op(L)], [0,1,op(L[2..-1])]
  else [0,op(L)]
  fi
end proc:
build:= proc(L) local i,k;
  k:= nops(L);
  add((F[i]+F[k+i])*L[i],i=1..k)
end proc:
R[2]:= [[0,1]]:
for i from 3 to 10 do R[i]:= map(ext,R[i-1]) od:
map(build, [seq(op(R[i]),i=2..10)]); # Robert Israel, Feb 19 2019

Reap[Do[ w = IntegerDigits[k, 2]; p = 1 + Flatten@ Position[ Reverse@ Join[w, w], 1]; If[ Min@ Differences@ p > 1, Sow@ Total@ Fibonacci@ p], {k, 2^10 - 1}]][[2, 1]] (* Giovanni Resta, May 13 2017 *)

a(A000045(n)) = A000045(n+1) + A000045(2n+1) for n >= 2. - Robert Israel, Feb 19 2019

A288252 Positive integers n such that the Fibonacci (or Zeckendorf) representation of n^2 is a palindrome.

Original entry on oeis.org

1, 2, 3, 8, 21, 38, 55, 80, 144, 168, 174, 195, 314, 377, 682, 987, 2584, 6360, 6765, 12238, 13301, 17711, 34985, 46368, 54096, 66483, 87849, 121393, 219602, 317811, 684704, 832040, 1486717, 2178309, 3325460, 3940598, 5702887, 6151102, 10008701, 14930352

Offset: 1

Jeffrey Shallit, Jun 07 2017

nonn

The sequence is infinite because F(2n)^2 = A049684(n) has Fibonacci (or Zeckendorf) representation (1000)^(n-1) 1.

			38 is in the sequence because 38^2 = 1444 has Fibonacci representation 101000101000101, which is a palindrome.

Giovanni Resta, Table of n, a(n) for n = 1..60

Cf. A014417, which explains Fibonacci representation. Cf. A094202.

for n from 1 do
    zeck := A014417(n^2) ;
    if isA002113(zeck) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jun 16 2017

a(35)-a(39) from Alois P. Heinz, Jun 14 2018

a(40) from Giovanni Resta, Jun 15 2018

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 *)

A356395 Nonnegative numbers k such that the negaFibonacci representation of k (A215022(k)) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 11, 14, 21, 24, 35, 40, 50, 55, 58, 66, 82, 90, 108, 118, 126, 144, 147, 176, 189, 205, 234, 247, 273, 286, 296, 325, 338, 364, 377, 380, 401, 443, 464, 511, 527, 548, 590, 611, 658, 684, 705, 752, 762, 783, 825, 846, 893, 919, 940, 987, 990

Offset: 1

Rémy Sigrist, Aug 05 2022

See A094202 and A356396 for similar sequences.

			The first terms are:
  n   a(n)  A215022(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3            101
   4     6          10001
   5     8          10101
   6    11        1001001
   7    14        1000001
   8    21        1010101
   9    24      100101001
  10    35      100000001

Cf. A094202, A215022, A356396.

is(n) = { my (v=0, neg=0, pos=0, f); for (e=0, oo, f=fibonacci(-1-e); if (f<0, neg+=f, pos+=f); if (neg <=n && n <= pos, while (n, if (f<0, neg-=f, pos-=f); if (neg > n || n > pos, v+=2^e; n-=f); f=fibonacci(-1-e--)); my (b=binary(v)); return (b==Vecrev(b)))) }

A356396 Nonnegative numbers k such that the negaFibonacci representation of -k (A215023(k)) is palindromic.

Original entry on oeis.org

0, 2, 7, 20, 26, 44, 54, 73, 112, 143, 159, 196, 212, 264, 290, 350, 376, 426, 492, 518, 568, 675, 756, 798, 905, 986, 1028, 1125, 1167, 1280, 1361, 1403, 1500, 1542, 1683, 1751, 1908, 1976, 2107, 2290, 2358, 2515, 2583, 2714, 2887, 2955, 3086, 3275, 3343

Offset: 1

Rémy Sigrist, Aug 05 2022

See A094202 and A356395 for similar sequences.

			The first terms are:
  n   a(n)  A215023(a(n))
  --  ----  -------------
   1     0              0
   2     2           1001
   3     7         100001
   4    20       10000001
   5    26       10100101
   6    44     1001001001
   7    54     1000000001
   8    73     1010000101
   9   112   100100001001
  10   143   100000000001

Cf. A094202, A215023, A356395.

is(n) = { my (v=0, neg=0, pos=0, f); n=-n; for (e=0, oo, f=fibonacci(-1-e); if (f<0, neg+=f, pos+=f); if (neg <=n && n <= pos, while (n, if (f<0, neg-=f, pos-=f); if (neg > n || n > pos, v+=2^e; n-=f); f=fibonacci(-1-e--)); my (b=binary(v)); return (b==Vecrev(b)))) }

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

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

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A286710 Numbers n whose Zeckendorf representation is of the form ww, for w a nonempty block of digits.

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A288252 Positive integers n such that the Fibonacci (or Zeckendorf) representation of n^2 is a palindrome.

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

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A356395 Nonnegative numbers k such that the negaFibonacci representation of k (A215022(k)) is palindromic.

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A356396 Nonnegative numbers k such that the negaFibonacci representation of -k (A215023(k)) is palindromic.

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A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

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

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