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

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

Author

Amiram Eldar, Jan 11 2020

Keywords

• nonn
• base

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

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

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