A095276 - OEIS

%I #25 Jun 27 2024 03:52:27

%S 1,3,1,1,2,1,3,2,3,1,1,3,3,1,1,2,1,3,1,1,3,1,1,2,1,3,2,3,1,1,2,1,3,1,

%T 1,2,1,3,2,3,1,1,3,3,1,1,2,1,3,2,3,1,1,2,1,3,2,3,1,1,3,3,1,1,2,1,3,1,

%U 1,3,1,1,2,1,3,2,3,1,1,3,3,1,1,2,1,3,2,3,1,1,3,3,1,1,2,1,3,1,1,3,1,1

%N Length of n-th run of identical symbols in A095076 and A095111.

%C Conjecture: it appears that the asymptotic frequencies of terms 1, 2 and 3 are 1/2, 1/(2*phi^2) and 1/(2*phi) respectively, where phi = (1+sqrt(5))/2 is the golden ratio. - _Vladimir Reshetnikov_, Mar 17 2022

%C For a proof of this conjecture see my link to a095276.pdf. - _Michel Dekking_, Jun 25 2024

%H Amiram Eldar, <a href="/A095276/b095276.txt">Table of n, a(n) for n = 1..10000</a>

%H Michel Dekking, <a href="/A095276/a095276.pdf">Proof of conjecture on frequencies</a>

%F (a(n)) is a morphic sequence. Let y = GDAEABFA... be the unique fixed point of the morphism rho given by rho(A) = B, rho(B) =C, rho(C) = F, rho(D) = EA, rho(E) = FA, rho(F) = GA, rho(G) = GDA on the alphabet  {A,B,C,D,E,F,G}. Then (a(n+1)) is the image of y under the morphism  A->11, B->21, C->32, D->23, E->33, F->3113, G->311213. - _Michel Dekking_, Jun 25 2024

%t Length /@ Split[Mod[DigitCount[Select[Range[0, 1500], BitAnd[#, 2 #] == 0 &], 2, 1], 2]] (* _Amiram Eldar_, Feb 07 2023 *)

%Y Partials sums: A095279.

%Y Cf. A001622, A026465, A095076, A095111.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 01 2004