A094202 -id:A094202 - cp's OEIS frontend

A348571 In Zeckendorf representation: integers that set a new record for the number of Reverse and Add steps (A349239) needed to reach a palindrome (A094202).

0, 2, 7, 20, 54, 63, 114, 1002, 1413, 3007, 4447, 35131, 599185, 2189416, 2738842, 3253273, 108250112

Offset: 1

A.H.M. Smeets, Oct 23 2021

Corresponding record values in A348572.
For Zeckendorf representation of numbers see A014417.
Lychrel numbers, as given in A348570, are excluded from this list because it is believed that those numbers never reach a palindrome.

			Trajectory of 20, i.e., 101010 in Zeckendorf representation:
       101010 + 010101      =      1010100
      1010100 + 0010101     =     10010010
     10010010 + 01001001    =    100100100
    100100100 + 001001001   =   1000010001
   1000010001 + 1000100001  =  10100000010
  10100000010 + 01000000101 = 100100001001, which is palindromic.
Due to the fact that any number smaller than 20 reaches a palindrome in fewer than 6 steps, 20 is a record-setting nonnegative integer.
The Lychrel numbers, as given in A348570, are excluded, because it is believed that those numbers never reach a palindromic number.

A.H.M. Smeets, Python program

Cf. A014417 (Zeckendorf digits), A349239 (reverse and add), A094202 (palindromes).

Cf. A348572 (number of steps), A348570 (Lychrels).

A331191 Numbers whose dual Zeckendorf representations (A104326) are palindromic.

Original entry on oeis.org

0, 1, 3, 4, 6, 11, 12, 16, 19, 22, 32, 33, 38, 42, 48, 53, 64, 71, 87, 88, 98, 106, 110, 118, 124, 134, 142, 148, 174, 194, 205, 231, 232, 245, 255, 271, 284, 288, 304, 317, 323, 336, 346, 362, 375, 402, 420, 462, 474, 516, 548, 566, 608, 609, 635, 656, 666, 687

Offset: 1

Amiram Eldar, Jan 11 2020

Pairs of numbers of the form {F(2*k-1)-2, F(2*k-1)-1}, for k >= 2, where F(k) is the k-th Fibonacci number, are consecutive terms in this sequence: {0, 1}, {3, 4}, {11, 12}, {32, 33}, ... - Amiram Eldar, Sep 03 2022

			4 is a term since its dual Zeckendorf representation, 101, is palindromic.

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

Cf. A000045, A002113, A006995, A014190, A094202, A104326.

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]]);
pals = Join[{{}}, Rest[Select[IntegerDigits[Range[0, 2^6 - 1], 2], SequenceCount[#, {0, 0}] == 0 &]]];
Union@Join[{0}, fib /@ Join[mirror[#, {}] & /@ (select[pals, 0]), mirror[#, {0}] & /@ (select[pals, 0]), mirror[#, {1}] & /@ pals]]

A351712 Numbers whose minimal (or greedy) Lucas representation (A130310) is palindromic.

Original entry on oeis.org

0, 2, 6, 9, 13, 20, 24, 31, 49, 56, 64, 78, 100, 125, 136, 150, 158, 169, 201, 237, 252, 324, 342, 364, 378, 396, 404, 422, 444, 523, 581, 606, 650, 708, 845, 874, 910, 932, 961, 975, 1004, 1040, 1048, 1077, 1113, 1135, 1164, 1366, 1460, 1500, 1572, 1666, 1692, 1786

Offset: 1

Amiram Eldar, Feb 17 2022

A000211(n) = Lucas(n) + 2 is a term for all n > 2, since the representation of Lucas(n) + 2 is 10...01 with n-1 0's between the two 1's.

			The first 10 terms are:
   n  a(n) A130310(a(n))
   ---------------------
   1   0               0
   2   2               1
   3   6            1001
   4   9           10001
   5  13          100001
   6  20         1000001
   7  24         1001001
   8  31        10000001
   9  49       100000001
  10  56       100010001

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A000211, A130310.
Subsequence of A054770.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351717.

lucasPalQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; PalindromeQ[IntegerDigits[Total[2^s], 2]]]; Select[Range[0, 2000], lucasPalQ]

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

Original entry on oeis.org

0, 2, 3, 5, 6, 10, 12, 14, 17, 20, 28, 30, 34, 36, 42, 46, 56, 61, 75, 77, 85, 92, 94, 101, 107, 115, 122, 128, 150, 166, 176, 198, 200, 211, 219, 233, 244, 246, 260, 271, 277, 288, 296, 310, 321, 345, 360, 396, 405, 441, 469, 484, 520, 522, 544, 562, 570, 588

Offset: 1

Amiram Eldar, Feb 17 2022

A001610(n) = Lucas(n+1) - 1 is a term for all n, since A001610(0) = 0 has the representation 0 and the representation of Lucas(n+1) - 1 is n 1's for n > 0.

			The first 10 terms are:
   n  a(n)  A130311(a(n))
   ----------------------
   1   0               0
   2   2               1
   3   3              11
   4   5             101
   5   6             111
   6  10            1111
   7  12           10101
   8  14           11011
   9  17           11111
  10  20          101101

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A001610, A130311.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712.

lazy = Select[IntegerDigits[Range[6000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Join[{0}, Position[s, _?PalindromeQ] // Flatten]

A352087 Numbers whose minimal (or greedy) tribonacci representation (A278038) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 8, 14, 18, 23, 25, 36, 40, 45, 52, 62, 71, 78, 82, 102, 110, 128, 148, 150, 163, 181, 198, 211, 229, 233, 246, 264, 275, 312, 326, 360, 397, 411, 426, 463, 477, 505, 529, 562, 593, 617, 650, 658, 682, 715, 746, 770, 781, 805, 838, 869, 893, 926, 928

Offset: 1

Amiram Eldar, Mar 04 2022

A000073(n) + 1 is a term for n>=4, since its minimal tribonacci representation is 10...01 with n-4 0's between the two 1's.

			The first 10 terms are:
   n  a(n)  A278038(a(n))
  -----------------------
   1   0                0
   2   1                1
   3   3               11
   4   5              101
   5   8             1001
   6  14            10001
   7  18            10101
   8  23            11011
   9  25           100001
  10  36           101101

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

Cf. A000073, A278038.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; PalindromeQ[FromDigits @ IntegerDigits[Total[2^(s - 1)], 2]]]; Select[Range[0, 1000], q]

A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 14, 18, 23, 27, 36, 40, 51, 52, 62, 69, 78, 88, 95, 102, 110, 130, 148, 156, 176, 181, 194, 211, 229, 242, 246, 264, 277, 294, 312, 325, 326, 363, 397, 411, 448, 463, 477, 514, 548, 562, 599, 617, 650, 674, 682, 715, 739, 770, 803, 827, 838, 862

Offset: 1

Amiram Eldar, Mar 05 2022

A027084(n) is a term since its maximal tribonacci representation is n-1 1's and no 0's.

The pairs {A008937(3*k+1)-1, A008937(3*k+1)} = {0, 1}, {7, 8}, {51, 52}, ... are consecutive terms in this sequence: the maximal tribonacci representation of A008937(3*k+1)-1 is 3*k 1's and no 0's (except for k=0 where the representation is 0), and the maximal tribonacci representation of A008937(3*k+1) is of the form 100100...1001 with k blocks of 100 followed by a 1 at the end.

			The first 10 terms are:
   n  a(n)  A352103(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    5             101
   5    7             111
   6    8            1001
   7   14            1111
   8   18           10101
   9   23           11011
  10   27           11111

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

Cf. A000073, A008937, A352103.
A027084 is a subsequence.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; q[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, True, PalindromeQ[FromDigits[v[[i[[1, 1]] ;; -1]]]]]]; Select[Range[0, 1000], q]

A095309 Numbers such that both their binary and Zeckendorf representations are palindromic.

Original entry on oeis.org

0, 1, 3, 9, 27, 33, 51, 127, 1755, 2805, 10437, 71377, 547233, 1007727, 2924109, 3358515, 3460299, 59768775, 977921175, 1022225871, 1769996491, 5606742245, 13759209651, 15569747991, 120793923335, 426202820195, 6287935078637, 21296868044633, 25068131362413

Offset: 1

Robert G. Wilson v, Jun 01 2004

Chai Wah Wu, Table of n, a(n) for n = 1..38

Intersection of A006995 and A094202.

fbz[n_] := Block[{k = Floor[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, a = {}}, While[k > 1, If[Fibonacci[k] <= t, t = t - Fibonacci[k]; AppendTo[a, 1], AppendTo[a, 0]]; k-- ]; a]; Do[b = IntegerDigits[2n + 1, 2]; If[b == Reverse[b], f = fbz[2n + 1]; If[f == Reverse[f], Print[2n + 1]]], {n, 0, 10^9}]

a(22)-a(29) from Chai Wah Wu, Jun 14 2018

a(1) = 0 added by Amiram Eldar, Jan 11 2020

A331891 Negabinary palindromes: nonnegative numbers whose negabinary expansion (A039724) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 17, 21, 23, 31, 43, 51, 57, 65, 77, 85, 87, 103, 127, 143, 155, 171, 195, 211, 217, 233, 257, 273, 285, 301, 325, 341, 343, 375, 423, 455, 479, 511, 559, 591, 603, 635, 683, 715, 739, 771, 819, 851, 857, 889, 937, 969, 993, 1025, 1073, 1105, 1117

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 its negabinary representation is 101 which is palindromic.

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

Cf. A002113, A006995, A014190, A039724, A094202, A331191.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 1200], PalindromeQ @ negabin[#] &]

A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 13, 20, 30, 35, 40, 44, 49, 71, 88, 102, 119, 170, 182, 194, 204, 216, 238, 242, 254, 266, 276, 288, 409, 450, 484, 525, 559, 580, 621, 655, 696, 986, 1015, 1044, 1068, 1097, 1150, 1160, 1189, 1218, 1242, 1271, 1334, 1363, 1392, 1396, 1425, 1454

Offset: 1

Amiram Eldar, Mar 12 2022

A052937(n) = A000129(n+1)+1 is a term for n>0, since its minimal Pell representation is 10...01 with n-1 0's between two 1's.
A048739 is a subsequence since these are repunit numbers in the minimal Pell representation.
A001109 is a subsequence. The minimal Pell representation of A001109(n), for n>1, is 1010...01, with n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A317204(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3             11
   4     6            101
   5     8            111
   6    13           1001
   7    20           1111
   8    30          10001
   9    35          10101
  10    40          10201

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

Cf. A000129, A317204.
Subsequences: A001109, A048739, A052937 \ {2}.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; PalindromeQ[IntegerDigits[Total[3^(s - 1)], 3]]]; Select[Range[0, 1500], q]

A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 20, 27, 40, 49, 54, 58, 63, 68, 88, 93, 119, 136, 150, 167, 221, 238, 288, 300, 310, 322, 334, 338, 360, 372, 382, 394, 406, 508, 530, 542, 696, 737, 771, 812, 833, 867, 908, 942, 983, 1242, 1276, 1317, 1392, 1681, 1710, 1734, 1763, 1792, 1802

Offset: 1

Amiram Eldar, Mar 12 2022

A000129(n) - 2 is a term for n > 1. The maximal Pell representations of these numbers are 0, 11, 121, 1221, 12221, ... (0 and A132583).
A048739 is a subsequence since these are the repunit numbers in the maximal Pell representation.
A065113 is a subsequence since the maximal Pell representation of A065113(n) is 2*n 2's.

			The first 10 terms are:
   n  a(n)  A352339(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    6              22
   5    8             111
   6   10             121
   7   20            1111
   8   27            1221
   9   40            2222
  10   49           11111

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

Cf. A000129, A048739, A065113, A132583, A352339.

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

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazy[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Select[Range[0, 2000], PalindromeQ[lazy[#]] &]

cp's OEIS Frontend

A348571 In Zeckendorf representation: integers that set a new record for the number of Reverse and Add steps (A349239) needed to reach a palindrome (A094202).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A331191 Numbers whose dual Zeckendorf representations (A104326) are palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A351712 Numbers whose minimal (or greedy) Lucas representation (A130310) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352087 Numbers whose minimal (or greedy) tribonacci representation (A278038) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A095309 Numbers such that both their binary and Zeckendorf representations are palindromic.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A331891 Negabinary palindromes: nonnegative numbers whose negabinary expansion (A039724) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

Views