A130823 Each odd number appears thrice.

1, 1, 1, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 13, 13, 13, 15, 15, 15, 17, 17, 17, 19, 19, 19, 21, 21, 21, 23, 23, 23, 25, 25, 25, 27, 27, 27, 29, 29, 29, 31, 31, 31, 33, 33, 33, 35, 35, 35, 37, 37, 37, 39, 39, 39, 41, 41, 41, 43, 43, 43, 45, 45, 45, 47, 47, 47, 49, 49

Offset

1,4

Comments

Partial sums of 1,0,0,2,0,0,2,0,0,2,0,0,... . - Emeric Deutsch, Jul 23 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

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

Formula

G.f.: x*(1 + x^3)/((1 - x)*(1 - x^3)). - Emeric Deutsch, Jul 23 2007

From Michael Somos, Aug 16 2007: (Start)

Euler transform of length 6 sequence [1, 0, 2, 0, 0, -1].

a(n + 3) = a(n) + 2.

a(n) = - a(1-n) for all n in Z. (End)

a(n) = -1 + Sum_{k=1..n} ((2/9)*((k mod 3)+4*((k+1) mod 3)-2*((k+2) mod 3))). - Paolo P. Lava, Aug 29 2007, Aug 22 2009

a(n) = floor((n-1)/3)*2 + 1. Michael Somos' formula suggested by Johannes W. Meijer. - Paolo P. Lava, Aug 21 2009

a(n) = floor((n-1)*(n+1)/3) - floor((n-2)*n/3). - Bruno Berselli, Mar 03 2017

a(n) = (6*n-3-4*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017

Example

G.f. = x + x^2 + x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 5*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + ...

Maple

G:=x*(1+x^3)/((1-x)*(1-x^3)): Gser:=series(G, x=0, 82): seq(coeff(Gser, x, n), n= 1..75); # Emeric Deutsch, Jul 23 2007

Mathematica

Flatten[Table[n, {n, 1, 49, 2}, {3}]] (* or *) LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 3}, 100] (* or *) Accumulate[PadRight[{1}, 100, {2, 0, 0}]] (* Harvey P. Dale, Apr 20 2015 *)

Prog

(PARI) {a(n) = (n-1)\3*2+1}; \\ Michael Somos, Aug 16 2007
(Magma) [Floor((n-1)/3)*2+1: n in [1..80]]; // Vincenzo Librandi, Aug 10 2011

Crossrefs

Sequence in context: A200266 A101290 A080605 * A101435 A077886 A096015

Adjacent sequences: A130820 A130821 A130822 * A130824 A130825 A130826

Keyword

nonn,easy

Author

Paul Curtz, Jul 17 2007

Extensions

More terms from Emeric Deutsch, Jul 23 2007

Status

approved