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A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.

%I A381580 #7 Feb 28 2025 12:08:12
%S A381580 0,1,2,4,9,12,15,18,22,33,44,56,64,72,80,88,96,104,112,120,128,136,
%T A381580 145,174,203,232,261,290,319,348,378,399,420,441,462,483,504,525,546,
%U A381580 567,588,609,630,651,672,693,714,735,756,777,798,819,840,861,882,903,924,945,966,988
%N A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.
%C A381580 The numbers of the form Fibonacci(2*k) + 1 (A055588) are all terms since A381579(A055588(0)) = 1, A381579(A055588(1)) = 2, and A381579(A055588(k)) = 10^(k-1)+1 (i.e., two 1's with k-2 0's between them) for k >= 2.
%H A381580 Amiram Eldar, <a href="/A381580/b381580.txt">Table of n, a(n) for n = 1..10000</a>
%e A381580 The first 10 terms are:
%e A381580    n  a(n) A381579(a(n))
%e A381580    ---------------------
%e A381580    1   0               0
%e A381580    2   1               1
%e A381580    3   2               2
%e A381580    4   4              11
%e A381580    5   9             101
%e A381580    6  12             111
%e A381580    7  15             121
%e A381580    8  18             202
%e A381580    9  22            1001
%e A381580   10  33            1111
%t A381580 f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2*10^(k-1); m -= 2*f[k], s += 10^(k-1); m -= f[k]]]; PalindromeQ[s]]; Select[Range[0, 1000], q]
%o A381580 (PARI) mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
%o A381580 isok(n) = {my(s = 0, m = n, k, d); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2*10^(k-1); m -= 2*f(k), s += 10^(k-1); m -= f(k))); d = digits(s); Vecrev(d) == d;}
%Y A381580 Cf. A000045, A381579.
%Y A381580 Subsequence: A055588.
%Y A381580 Similar sequences: A002113, A006995, A094202, A331191.
%K A381580 nonn,easy,base
%O A381580 1,3
%A A381580 _Amiram Eldar_, Feb 28 2025