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
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
-
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[#] &]]
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
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
-
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[#] &]]
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
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
-
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[#] &]]
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