A095276 Length of n-th run of identical symbols in A095076 and A095111.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jun 01 2004

Keywords

• nonn

Comments

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

For a proof of this conjecture see my link to a095276.pdf. - Michel Dekking, Jun 25 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Michel Dekking, Proof of conjecture on frequencies

Crossrefs

Partials sums: A095279.

Cf. A001622, A026465, A095076, A095111.

Programs

Mathematica

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

Formula

(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