A331191 -id:A331191 - cp's OEIS frontend

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

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]

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[#]] &]

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A331193 Numbers whose binary and dual Zeckendorf representations are both palindromic.

Original entry on oeis.org

0, 1, 3, 33, 231, 255, 891, 3687, 21477, 1216041, 5360069, 418964451, 443750859, 1445812789, 23577810421, 25474675645, 154292473329, 1904542477755, 1925488579591, 9617724354513, 16654480398927, 169215938357145, 2563713753111945, 3408057776446851, 4019397080882727

Offset: 1

Amiram Eldar, Jan 11 2020

			3 is a term since both its binary and dual Zeckendorf representations are 11 which is palindromic.
33 is a term since its binary representation, 100001, and its dual Zeckendorf representation, 1010101, are both palindromic.

Chai Wah Wu, Table of n, a(n) for n = 1..29

Intersection of A006995 and A331191.

Cf. A095309, 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 /@ FromDigits /@ Tuples[{0, 1}, 22], SequenceCount[#, {0, 0}] == 0 &]]];
dualZeckPals = Union @ Join[{0}, fib /@ Join[mirror[#, {}] & /@ (select[pals, 0]), mirror[#, {0}] & /@ (select[pals, 0]), mirror[#, {1}] & /@ pals]];
binPalQ[n_] := PalindromeQ@IntegerDigits[n, 2]; Select[dualZeckPals, binPalQ]

a(18)-a(22) from Chai Wah Wu, Jan 12 2020

a(23)-a(25) from Chai Wah Wu, Jan 13 2020

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015

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 the negabinary representation of -5 is 1111 which is palindromic.

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

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

cp's OEIS Frontend

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

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

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

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

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

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

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

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A342725 Numbers that are palindromic in base i-1.

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