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 1-7 of 7 results.

A331894 Positive numbers such that both their binary and negabinary representations are palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 17, 21, 31, 51, 65, 85, 127, 195, 257, 273, 325, 341, 455, 511, 771, 819, 1025, 1105, 1285, 1365, 1799, 2047, 3075, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 7175, 7967, 8191, 12291, 12483, 13107, 16385, 16705, 17425, 17745, 20485, 20805
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2020

Keywords

Comments

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

Examples

			7 is a term since the binary representation of 7, 111, and the negabinary representation of 7, 11011, are both palindromic.

Links

Crossrefs

Intersection of A006995 and A331891.
Cf. A039724, A095309.

Programs

  • Mathematica
    negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 2*10^4], And @@ (PalindromeQ /@ {IntegerDigits[#, 2], negabin[#]}) &]

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

Views

Author

Amiram Eldar, Jan 11 2020

Keywords

Examples

			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.

Links

Crossrefs

Intersection of A006995 and A331191.
Cf. A095309, A104326.

Programs

  • Mathematica
    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]

Extensions

a(18)-a(22) from Chai Wah Wu, Jan 12 2020
a(23)-a(25) from Chai Wah Wu, Jan 13 2020

A351713 Numbers whose binary and minimal Lucas representations are both palindromic.

Original entry on oeis.org

0, 9, 31, 975, 297097, 816867, 4148165871, 152488124529, 1632977901693, 11162529166917, 11925833175477, 3047549778123957, 3894487365191355, 8920885515768255
Offset: 1

Views

Author

Amiram Eldar, Feb 17 2022

Keywords

Examples

			   n    a(n)       A007088(a(n))                A130310(a(n))
   ----------------------------------------------------------
   1       0                   0                            0
   2       9                1001                        10001
   3      31               11111                     10000001
   4     975          1111001111              100010000010001
   5  297097 1001000100010001001  100001000000101000000100001

Links

Crossrefs

Intersection of A006995 and A351712.
Subsequence of A054770.
Cf. A007088, A130310.
Similar sequences: A095309, A331193, A331894, A351718.

Programs

  • Mathematica
    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]]]; Join[{0}, Select[Range[1, 10^6, 2], PalindromeQ[IntegerDigits[#, 2]] && lucasPalQ[#] &]]

A329459 Numbers whose ternary and Zeckendorf representations are both palindromic.

Original entry on oeis.org

0, 1, 4, 56, 80, 203, 572, 847, 1402, 93496, 128180, 431060, 467852, 1465676, 7742920, 8727388, 8923840, 9582707, 18245944, 18304588, 25154692, 27262924, 115404434, 209060644, 763786258, 860973806, 2042328148, 4719261289, 5236838932, 18202403140, 42897493894, 77310551669
Offset: 1

Views

Author

Alex Ratushnyak, Nov 13 2019

Keywords

Comments

Intersection of A014190 and A094202.

Links

Crossrefs

Cf. A014190, A094202, A060792, A095309.

Extensions

a(24)-a(32) from Daniel Suteu, Nov 16 2019

A352088 Numbers whose binary and minimal tribonacci representations are both palindromic.

Original entry on oeis.org

0, 1, 3, 5, 45, 2193, 7671, 35889, 53835, 74825, 3026205, 31953871, 86582437, 117169915, 128873391, 701373669, 868430067, 15262037703, 45305389845, 104484026691, 614071181169, 14894476590363, 24382189266573, 86808432666553, 869188423288227, 1352557858988953
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2022

Keywords

Examples

			The first 5 terms are:
  n  a(n)  A007088(a(n))  A278038(a(n))
  -------------------------------------
  1     0              0              0
  2     1              1              1
  3     3             11             11
  4     5            101            101
  5    45         101101        1000001

Crossrefs

Intersection of A006995 and A352087.
Cf. A007088, A278038.
Similar sequences: A095309, A331193, A331894, A351713, A351718.

Programs

  • Mathematica
    t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; tribPalQ[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]]]; Join[{0}, Select[Range[1, 10^5, 2], PalindromeQ[IntegerDigits[#, 2]] && tribPalQ[#] &]]

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

Views

Author

Amiram Eldar, Jan 11 2020

Keywords

Comments

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.

Examples

			6 is a term since its Zeckendorf representation, 1001, and its dual Zeckendorf representation, 111, are both palindromic.

Crossrefs

Intersection of A094202 and A331191.
Cf. A000045, A000071, A002113, A006995, A014190, A027941, A048268, A060792, A095309, A104326, A329459.

Programs

  • Mathematica
    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]

A352106 Numbers whose binary and maximal tribonacci representations are both palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 27, 51, 325, 2193, 3735, 23709, 35889, 53835, 589833, 1294265, 17291201, 80719769, 1274288105, 23157444917, 23635236877, 230684552043, 1218891196337, 1722894010643, 2544113575977, 93096801594005, 175482093541881, 256924005422487, 372295593308821
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2022

Keywords

Examples

			The first 5 terms are:
   n  a(n)  A007088(a(n))  A352103(a(n))
   -  ----  -------------  -------------
   1     0              0              0
   2     1              1              1
   3     3             11             11
   4     5            101            101
   5     7            111            111
   6    27          11011          11111
   7    51         110011         111111
   8   325      101000101      111111111
   9  2193   100010010001  1001101011001
  10  3735   111010010111  1111111111111

Crossrefs

Intersection of A006995 and A352105.
Cf. A007088, A352103.
Similar sequences: A095309, A331193, A331894, A351713, A351718, A352088.

Programs

  • Mathematica
    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]]; lazyTribPalQ[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]]]]]]; Join[{0}, Select[Range[1, 10^5, 2], PalindromeQ[IntegerDigits[#, 2]] && lazyTribPalQ[#] &]]
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.