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A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.

%I A364005 #11 Jul 01 2023 09:20:22
%S A364005 0,1,2,5,7,10,13,15,23,28,34,36,52,57,65,75,81,89,91,117,128,146,159,
%T A364005 175,185,198,204,217,233,235,277,295,327,369,379,400,426,442,463,473,
%U A364005 494,520,526,547,573,589,610,612,680,709,761,829,848,916,945,989,1023
%N A364005 Numbers whose Wythoff representation (A189921, A317208) is palindromic.
%C A364005 Includes all the odd-indexed Fibonacci numbers (A001519), since the Wythoff representation of Fibonacci(1) is 1 and the Wythoff representation of Fibonacci(2*n+1), for n >= 1, is n 0's.
%C A364005 A157725(n) = Fibonacci(n) + 2 is a term for n >= 4, since its Wythoff representation is n-4 1's between 2 0's.
%C A364005 A232970 is a subsequence since the Wythoff representation of A232970(n) = (Fibonacci(3*n+1) + 1)/2 is n 0's and n-1 1's interleaved.
%H A364005 Amiram Eldar, <a href="/A364005/b364005.txt">Table of n, a(n) for n = 1..10000</a>
%e A364005 The first 10 terms are:
%e A364005    n  a(n)  A317208(a(n))
%e A364005   --  ----  -------------
%e A364005    1     0              0
%e A364005    2     1              1
%e A364005    3     2              2
%e A364005    4     5             22
%e A364005    5     7            212
%e A364005    6    10           2112
%e A364005    7    13            222
%e A364005    8    15          21112
%e A364005    9    23         211112
%e A364005   10    28          21212
%t A364005 z[n_] := Floor[(n + 1)*GoldenRatio] - n - 1; h[n_] := z[n] - z[n - 1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; w[0] = {0}; Select[Range[0, 1000], PalindromeQ[w[#]] &]
%Y A364005 Cf. A001519, A157725, A189921, A232970, A317208.
%Y A364005 Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A352507.
%K A364005 nonn,base
%O A364005 1,3
%A A364005 _Amiram Eldar_, Jul 01 2023