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%I #9 Mar 14 2022 02:42:03
%S 0,1,3,6,8,10,20,27,40,49,54,58,63,68,88,93,119,136,150,167,221,238,
%T 288,300,310,322,334,338,360,372,382,394,406,508,530,542,696,737,771,
%U 812,833,867,908,942,983,1242,1276,1317,1392,1681,1710,1734,1763,1792,1802
%N Numbers whose maximal Pell representation (A352339) is palindromic.
%C 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).
%C A048739 is a subsequence since these are the repunit numbers in the maximal Pell representation.
%C A065113 is a subsequence since the maximal Pell representation of A065113(n) is 2*n 2's.
%H Amiram Eldar, <a href="/A352341/b352341.txt">Table of n, a(n) for n = 1..10000</a>
%e The first 10 terms are:
%e n a(n) A352339(a(n))
%e -- ---- -------------
%e 1 0 0
%e 2 1 1
%e 3 3 11
%e 4 6 22
%e 5 8 111
%e 6 10 121
%e 7 20 1111
%e 8 27 1221
%e 9 40 2222
%e 10 49 11111
%t 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[#]] &]
%Y Cf. A000129, A048739, A065113, A132583, A352339.
%Y Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319.
%K nonn,base
%O 1,3
%A _Amiram Eldar_, Mar 12 2022
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