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A215245 a(n) = minimal value of A215244(k) for 2^n <= k < 2^(n+1).

%I A215245 #50 Oct 28 2019 20:05:19
%S A215245 1,1,2,3,4,6,9,13,20,29,42,65,95,136,212,308,444,687,1005,1439,2242,
%T A215245 3257,4696,7266,10629,15219
%N A215245 a(n) = minimal value of A215244(k) for 2^n <= k < 2^(n+1).
%C A215245 The initial terms roughly satisfy a(n) approx.= a(n-1)+a(n-3), which leads to the guess that perhaps a(n) ~ 1.4655^n, from the real zero of x^3-x-1. - _N. J. A. Sloane_, Aug 08 2012
%H A215245 Giovanni Resta, <a href="/A215245/a215245.txt">Examples of words attaining the minimal value for n = 0..25</a>
%e A215245 The values of A215244(k) for k=8 through 15 are (4, 3, 3, 3, 4, 3, 4, 8), with minimal value a(3) = 3.
%p A215245 A215245 := proc(n)
%p A215245     local a,k ;
%p A215245     a := A215244(2^n) ;
%p A215245     for k from 2^n+1 to 2^(n+1)-1 do
%p A215245         a := min(a,A215244(k)) ;
%p A215245     end do:
%p A215245     a ;
%p A215245 end proc: # _R. J. Mathar_, Aug 07 2012
%t A215245 palQ[L_] := SameQ[L, Reverse[L]];
%t A215245 b[L_] := b[L] = Module[{a = palQ[L] // Boole, c}, For[c = 1, c < Length[L], c++, If[palQ[L[[;; c]]], a = a + b[L[[c+1 ;;]]]]]; a];
%t A215245 a215244[n_] := If[n == 1, 1, b[IntegerDigits[n, 2]]];
%t A215245 a215245[n_] := Module[{a, k}, a = a215244[2^n]; For[k = 2^n+1, k <= 2^(n+1) - 1, k++, a = Min[a, a215244[k]]]; a];
%t A215245 a215245 /@ Range[0, 20] (* _Jean-François Alcover_, Oct 28 2019 *)
%Y A215245 Cf. A215244, A215246, A215253, A215254. A215255 gives an upper bound.
%K A215245 nonn,more,base
%O A215245 0,3
%A A215245 _N. J. A. Sloane_, Aug 07 2012
%E A215245 a(10)-a(13) from _R. J. Mathar_, Aug 07 2012
%E A215245 a(14)-a(17) from _N. J. A. Sloane_, Aug 08 2012, using Mathar's Maple code.
%E A215245 a(18)-a(25) from _Giovanni Resta_, Mar 19 2013