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A116522
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a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,..., a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,....
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7
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0, 1, 7, 13, 49, 55, 91, 127, 343, 349, 385, 421, 637, 673, 889, 1105, 2401, 2407, 2443, 2479, 2695, 2731, 2947, 3163, 4459, 4495, 4711, 4927, 6223, 6439, 7735, 9031, 16807, 16813, 16849, 16885, 17101, 17137, 17353, 17569, 18865, 18901, 19117, 19333
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OFFSET
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0,3
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COMMENTS
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The Harborth: f(2^k) = 3^k suggests that a family of sequences of the form: f(2^k) = prime(n)^k.
Let M = the production matrix below. Then lim_{k->infinity} M^k generates the sequence with offset 1 by extracting the left-shifted vector.
1, 0, 0, 0, 0, ...
7, 0, 0, 0, 0, ...
6, 1, 0, 0, 0, ...
0, 7, 0, 0, 0, ...
0, 6, 1, 0, 0, ...
0, 0, 7, 0, 0, ...
0, 0, 6, 1, 0, ...
...
The sequence divided by its aerated variant is (1, 7, 6, 0, 0, 0, ...). (End)
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LINKS
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FORMULA
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G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + 7x + 6x^2).
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MAPLE
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a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 7*a(n/2) else 6*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..47);
# second Maple program:
b:= proc(n) option remember; `if`(n<0, 0,
b(n-1)+x^add(i, i=Bits[Split](n)))
end:
a:= n-> subs(x=6, b(n-1)):
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MATHEMATICA
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b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 7*b[n/2]; b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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