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

1,2

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)

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

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

Example

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

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 *)

Prog

(PARI) baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x%b; x\=b; e+=d*f; f*=10); return(e) } { for (n=1, 1000, a=baseE(n, 4); while (a%10 == 0, a\=10); write("b065882.txt", n, " ", a%10) ) } \\ Harry J. Smith, Nov 03 2009
(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

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.

Sequence in context: A114280 A123564 A036466 * A276327 A007884 A190593

Adjacent sequences: A065879 A065880 A065881 * A065883 A065884 A065885

Keyword

base,easy,nonn

Author

Henry Bottomley, Nov 26 2001

Status

approved