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A301728
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a(0)=1; thereafter, a(n) = 2n-1 if n == 0 (mod 3), (5n+1)/3 if n == 1 (mod 3), (5n+2)/3 if n == 2 (mod 3).
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1
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1, 2, 4, 5, 7, 9, 11, 12, 14, 17, 17, 19, 23, 22, 24, 29, 27, 29, 35, 32, 34, 41, 37, 39, 47, 42, 44, 53, 47, 49, 59, 52, 54, 65, 57, 59, 71, 62, 64, 77, 67, 69, 83, 72, 74, 89, 77, 79, 95, 82, 84, 101, 87, 89, 107, 92, 94, 113, 97, 99, 119, 102, 104, 125
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OFFSET
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0,2
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REFERENCES
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A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p4".
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LINKS
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FORMULA
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G.f.: -(-2*x^6-x^5-3*x^4-3*x^3-4*x^2-2*x-1)/(x^6-2*x^3+1).
Recurrence (there are several equivalent versions):
-n a(n) - 2 a(n + 1) - a(n + 2) + (n - 3) a(n + 3) + 2 a(n + 4) + a(n + 5) = 0, with a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 5, a(4) = 7.
a(n) = 2*a(n-3) - a(n-6) for n > 6. - Chai Wah Wu, Mar 30 2018
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MAPLE
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f:= proc(n) if n=0 then 1
elif (n mod 3) = 0 then 2*n-1
elif (n mod 3) = 1 then (5*n+1)/3
else (5*n+2)/3; fi;
end;
[seq(f(n), n=0..70)];
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MATHEMATICA
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Join[{1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {2, 4, 5, 7, 9, 11}, 100]] (* Vincenzo Librandi, Mar 31 2018 *)
nxt[{n_, a_}]:={n+1, Which[Divisible[n+1, 3], 2n+1, Mod[n+1, 3]==1, (5n+6)/3, True, (5n+7)/3]}; NestList[nxt, {0, 1}, 70][[All, 2]] (* Harvey P. Dale, Sep 08 2021 *)
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PROG
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(Magma) I:=[1, 2, 4, 5, 7, 9, 11]; [n le 7 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Mar 31 2018
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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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