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%I #8 Jul 26 2024 08:57:35
%S 5,6,1931,4127,26584
%N Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of terms in their Zeckendorf representation (A007895).
%C Numbers k such that A020908(k) = A020908(k+1) = A020908(k+2).
%C The corresponding values of A020908(k) are 3, 3, 763, 1660, 10596, ... .
%C a(6) > 10^5, if it exists.
%e 5 is a term since A020908(5) = A020908(6) = A020908(7) = 3.
%e 763 is a term since A020908(1931) = A020908(1932) = A020908(1933) = 763.
%t z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* _Alonso del Arte_ at A007895 *)
%t s[n_] := s[n] = z[2^n]; Select[Range[0, 4200], s[#] == s[# + 1] == s[# + 2] &]
%o (PARI) A007895(n)=if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s); \\ _Charles R Greathouse IV_ at A007895
%o lista(kmax) = {my(z1 = A007895(1), z2 = A007895(2), z3); for(k = 2, kmax, z3 = A007895(2^k); if(z1 == z2 && z2 == z3, print1(k-2 , ", ")); z1 = z2; z2 = z3);}
%Y Cf. A007895, A020908, A353987.
%Y Subsequence of A374960.
%K nonn,base,hard,more
%O 1,1
%A _Amiram Eldar_, Jul 25 2024