A194960 a(n) = floor((n+2)/3) + ((n-1) mod 3).

1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25, 24, 25, 26, 25, 26

Offset

1,2

Comments

The sequence is formed by concatenating triples of the form (n, n+1, n+2) for n>=1. See A194961 and A194962 for the associated fractalization and interspersion. The sequence can be obtained from A008611 by deleting its first four terms.

The sequence contains every positive integer n exactly min(n,3) times. - Wesley Ivan Hurt, Dec 17 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Formula

From R. J. Mathar, Sep 07 2011: (Start)

a(n) = ((-1)^n*A130772(n) + n + 4)/3.

G.f.: x*(1 + x + x^2 - 2*x^3)/((1+x+x^2)*(1-x)^2). (End)

a(n) = A006446(n)/floor(sqrt(A006446(n))). - Benoit Cloitre, Jan 15 2012

a(n) = a(n-1) + a(n-3) - a(n-4). - Vincenzo Librandi, Dec 17 2013

a(n) = a(n-3) + 1, n >= 1, with input a(-2) = 0, a(-1) = 1 and a(0) = 2. Proof trivial. a(n) = A008611(n+3), n >= -2. See the first comment above. - Wolfdieter Lang, May 06 2017

From Guenther Schrack, Nov 09 2020: (Start)

a(n) = n - 2*floor((n-1)/3).

a(n) = (n + 2 + 2*((n-1) mod 3))/3.

a(n) = (3*n + 12 + 2*(w^(2*n)*(1 - w) + w^n*(2 + w)))/9, where w = (-1 + sqrt(-3))/2.

a(n) = (n + 4 + 2*A049347(n))/3.

a(n) = (2*n + 3 - A330396(n-1))/3. (End)

a(n) = (n + 4 - 2*A010892(2*n+4))/3. - G. C. Greubel, Oct 23 2022

Maple

A194960:=n->floor((n+2)/3)+((n-1) mod 3); seq(A194960(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2013

Mathematica

(* First program *)
p[n_]:= Floor[(n+2)/3] + Mod[n-1, 3]
Table[p[n], {n, 1, 90}]  (* A194960 *)
g[1] = {1}; g[n_]:= Insert[g[n-1], n, p[n]]
f[1] = g[1]; f[n_]:= Join[f[n-1], g[n]]
f[20]  (* A194961 *)
row[n_]:= Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_]:= Part[row[n], k];
w = Flatten[Table[v[k, n-k+1], {n, 1, 13}, {k, 1, n}]]  (* A194962 *)
q[n_]:= Position[w, n];
Flatten[Table[q[n], {n, 1, 80}]]  (* A194963 *)
(* Other programs *)
CoefficientList[Series[(1 +x +x^2 -2 x^3)/((1+x+x^2) (1-x)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 17 2013 *)
Table[(n+4 -2*ChebyshevU[2*n+4, 1/2])/3, {n, 80}] (* G. C. Greubel, Oct 23 2022 *)

Prog

(Magma) I:=[1, 2, 3, 2]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..100]]; // Vincenzo Librandi, Dec 17 2013
(PARI) a(n)=(n+2)\3 + (n-1)%3 \\ Charles R Greathouse IV, Sep 02 2015
(SageMath) [(n+4 - 2*chebyshev_U(2*n+4, 1/2))/3 for n in (1..80)] # G. C. Greubel, Oct 23 2022

Crossrefs

Cf. A006446, A008611, A010892, A047878, A049347, A130772, A194961, A194962, A194963, A330396.

Sequence in context: A045781 A045670 A363678 * A111439 A020986 A326820

Adjacent sequences: A194957 A194958 A194959 * A194961 A194962 A194963

Keyword

nonn,easy

Author

Clark Kimberling, Sep 06 2011

Status

approved