A301728 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).

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

Offset

0,2

References

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".

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Formula

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

Maple

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)];

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A093848 A049039 A325101 * A005152 A060831 A073727

Adjacent sequences: A301725 A301726 A301727 * A301729 A301730 A301731

Keyword

nonn

Author

N. J. A. Sloane, Mar 30 2018

Status

approved