A065882 Ultimate modulo 4: right-hand nonzero digit of n when written in base 4.

1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1

Offset: 1

Author

Henry Bottomley, Nov 26 2001

Keywords

• base
• easy
• nonn

Comments

From Bradley Klee, Sep 12 2015: (Start)

In some guise, this sequence is a linear encoding of the three fixed-point half-hex tilings (cf. Baake & Grimm, Frettlöh). Applying a permutation, morphism x -> 123x becomes x -> x123, which has three fixed points. Applying a partition, morphism x -> x123 becomes x ->{{3,2},{x,1}} or

3 2 3 2

3 1 2 1

3 2 3 2 3 2

x -> x 1 -> x 1 1 1 -> etc.,

which is the substitution rule for the half-hex tiling when the numbers 1,2,3 determine the direction of a dissecting diameter inscribed on each hexagon.

(End)

Examples

			a(7)=3 and a(112)=3, since 7 is written in base 4 as 13 and 112 as 1300.
		

References

• M. Baake and U. Grimm, Aperiodic Order Vol. 1, Cambridge University Press, 2013, page 205.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• D. Frettlöh, Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Dissertation, Universität Dortmund, 2002.
• Index entries for sequences related to final digits of numbers
• Index entries for sequences that are fixed points of mappings

Crossrefs

In base 2 this is A000012, base 3 A060236 and base 10 A065881.

Defining relations for g.f. similar to A014577.

Cf. A010873, A037898, A065883, A190593.

Programs

Maple

f:= proc(n)
local x:=n;
   while x mod 4 = 0 do x:= x/4 od:
   x mod 4;
end proc;
map(f, [$1..100]); # Robert Israel, Jan 05 2016

Mathematica

Nest[ Flatten[ # /. {1 -> {1, 2, 3, 1}, 2 -> {1, 2, 3, 2}, 3 -> {1, 2, 3, 3}}] &, {1}, 4] (* Robert G. Wilson v, May 07 2005 *)
b[n_] := CoefficientList[Series[
    With[{f0 = (x + 2 x^2 + 3 x^3)/(1 - x^4)},
     Nest[ (# /. x -> x^4) + f0 &, f0, Ceiling[Log[4, n/3]]]],
{x, 0, n}], x][[2 ;; -1]]; b[100](* Bradley Klee, Sep 12 2015 *)
Table[Mod[n/4^IntegerExponent[n, 4], 4], {n, 1, 120}] (* Clark Kimberling, Oct 19 2016 *)

PARI

a(n) = (n/4^valuation(n,4))%4; \\ Joerg Arndt, Sep 13 2015

Python

def A065882(n): return (n>>((~n & n-1).bit_length()&-2))&3 # Chai Wah Wu, Aug 21 2023

Formula

If n mod 4 = 0 then a(n) = a(n/4), otherwise a(n) = n mod 4. a(n) = A065883(n) mod 4.

Fixed point of the morphism: 1 ->1231, 2 ->1232, 3 ->1233, starting from a(1) = 1. Sequence read mod 2 gives A035263. a(n) = A007913(n) mod 4. - Philippe Deléham, Mar 28 2004

G.f. g(x) satisfies g(x) = g(x^4) + (x + 2 x^2 + 3 x^3)/(1 - x^4). - Bradley Klee, Sep 12 2015