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 A065882 Ultimate modulo 4: right-hand nonzero digit of n when written in base 4. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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(* 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 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: A308184 A114280 A123564 * A276327 A007884 A190593 Adjacent sequences:  A065879 A065880 A065881 * A065883 A065884 A065885 KEYWORD base,nonn AUTHOR Henry Bottomley, Nov 26 2001 STATUS approved

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Last modified November 12 22:16 EST 2019. Contains 329079 sequences. (Running on oeis4.)