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%I #12 Aug 10 2015 04:36:07
%S 0,1,2,1,1,0,3,0,1,0,3,3,2,3,0,0,1,0,3,0,0,1,2,1,0,1,2,2,3,2,1,1,1,0,
%T 3,0,0,1,2,1,0,1,2,2,3,2,1,2,2,3,0,3,3,2,1,2,3,2,1,1,0,1,2,1,1,0,3,0,
%U 0,1,2,1,0,1,2,2,3,2,1,1,0,1,2,1,1,0,3,0,1,0,3,3,2,3,0,0,0,1,2,1,1,0
%N The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.
%C Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).
%H A. Karttunen, <a href="/A163540/b163540.txt">Table of n, a(n) for n = 1..65536</a>
%F a(n) = A010873(A163538(n)+A163539(n)+abs(A163539(n))+3).
%t HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
%t R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
%t R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
%t L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
%t F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
%t F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
%t a[1] = F[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] :=
%t Part[Flatten[a[(n + 16 - #)/16] /. HC /. HC], #]) &, Range[16]];
%t Part[FoldList[Mod[Plus[#1, #2], 4] &, 0,
%t a[#] & /@ Range[4^4] /. {F[n_] :> 0, L[n_] :> 1, R[n_] :> -1}],
%t 2 ;; -1] (* _Bradley Klee_, Aug 07 2015 *)
%o (Scheme:) (define (A163540 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163539 n))) 4))
%Y a(n) = A163540(A008598(n)) = A004442(A163541(n)). See also A163542.
%K nonn
%O 1,3
%A _Antti Karttunen_, Aug 01 2009
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