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

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

Author

Amiram Eldar, Mar 12 2022

Keywords

• nonn
• base

Comments

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.

Examples

			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
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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[#]] &]