A330037 The sum of digits function modulo 2 of the natural numbers in base phi.

0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset

0

Comments

This sequence is a morphic sequence, i.e., a letter-to-letter image of a fixed point of a morphism.

Let the morphism tau on the alphabet A:={1,2,...,8} be defined by

tau(1) = 12, tau(2) = 312, tau(3) = 47, tau(4) = 8312,

tau(5) = 56, tau(6) = 756, tau(7) = 83, tau(8) = 4756.

Let lambda: A* -> {0,1} be the letter-to-letter morphism given by

lambda(1) = lambda(3) = lambda(6) = lambda(8) = 0;

lambda(2) = lambda(4) = lambda(5) = lambda(7) = 1.

Then a(n) = lambda(x(n)), where x(0)x(1)... = 123124712... is the fixed point of tau starting with 1. For a proof, see "The sum of digits function of the base phi expansion of the natural numbers".

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

F. M. Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.

Formula

a(n) = A055778(n) modulo 2.

Example

In base phi: 2 = 10.01, so a(2)=0; 3 = 100.01 so a(3)=0.

Crossrefs

See A130600 for the integers written in base phi, with the "decimal point" omitted. See A105424 for the part of n in base phi left of the decimal point.

Cf. A095076.

Sequence in context: A330731 A138710 A356556 * A255817 A071674 A179829

Adjacent sequences: A330034 A330035 A330036 * A330038 A330039 A330040

Keyword

nonn

Author

Michel Dekking, Nov 28 2019

Status

approved