cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Jul 01 2023

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A001519, A157725, A189921, A232970, A317208.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A352507.

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Jul 14 2023

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A001045, A002450, A003159, A020988, A280049.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Programs

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

Views

Author

Amiram Eldar, Jun 10 2022

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Jul 07 2023

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A000045, A200648, A200649, A200650, A200651, A200714, A364121.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Programs

  • Mathematica
    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]
  • PARI
    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

Views

Author

Amiram Eldar, Jul 21 2023

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A001045, A014825, A084639, A128209, A265747.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A364214.

Programs

  • Mathematica
    palJacobQ[n_] := PalindromeQ[A265747[n]]; Select[Range[0, 1000], palJacobQ] (* using A265747[n] *)
  • PARI
    is(n) = {my(dig = digits(A265747(n))); dig == Vecrev(dig);} \\ using A265747(n)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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