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

1,2

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

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

Mathematica

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

Crossrefs

Partials sums: A095279.

Cf. A001622, A026465, A095076, A095111.

Sequence in context: A334006 A364683 A270572 * A246457 A089338 A356400

Adjacent sequences: A095273 A095274 A095275 * A095277 A095278 A095279

Keyword

nonn

Author

Antti Karttunen, Jun 01 2004

Status

approved