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A114438
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Number of Barlow packings that repeat after n (or a divisor of n) layers.
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5
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0, 1, 1, 2, 1, 4, 3, 8, 8, 18, 21, 48, 63, 133, 205, 412, 685, 1354, 2385, 4644, 8496, 16431, 30735, 59344, 112531, 217246, 415628, 803210, 1545463, 2991192, 5778267, 11201884, 21702708, 42141576, 81830748, 159140896, 309590883, 602938099, 1174779397, 2290920128
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OFFSET
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1,4
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COMMENTS
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See A011768 for the number of Barlow packings that repeat after exactly n layers.
Like A056353 but with additional restriction that adjacent beads must have different colors.
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LINKS
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MAPLE
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with(numtheory); read transforms; M:=500;
A:=proc(N, d) if d mod 3 = 0 then 2^(N/d) else (1/3)*(2^(N/d)+2*cos(Pi*N/d)); fi; end;
E:=proc(N) if N mod 2 = 0 then N*2^(N/2) + add( did(N/2, d)*phi(2*d)*2^(N/(2*d)), d=1..N/2) else (N/3)*(2^((N+1)/2)+2*cos(Pi*(N+1)/2)); fi; end;
PP:=proc(N) (1/(4*N))*(add(did(N, d)*phi(d)*A(N, d), d=1..N)+E(N)); end; for N from 1 to M do lprint(N, PP(N)); od: # N. J. A. Sloane, Aug 10 2006
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MATHEMATICA
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M = 40;
did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])];
EE[n_] := If[Mod[n, 2] == 0, n 2^(n/2) + Sum[did[n/2, d] EulerPhi[2d] 2^(n/(2d)), {d, 1, n/2}], (n/3)(2^((n+1)/2) + 2 Cos[Pi(n+1)/2])];
a[n_] := (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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