A113655 Reverse blocks of three in the sequence of natural numbers.

3, 2, 1, 6, 5, 4, 9, 8, 7, 12, 11, 10, 15, 14, 13, 18, 17, 16, 21, 20, 19, 24, 23, 22, 27, 26, 25, 30, 29, 28, 33, 32, 31, 36, 35, 34, 39, 38, 37, 42, 41, 40, 45, 44, 43, 48, 47, 46, 51, 50, 49, 54, 53, 52, 57, 56, 55, 60, 59, 58, 63, 62, 61, 66, 65, 64, 69, 68, 67, 72, 71, 70

Offset: 1

Author

Parag D. Mehta (pmehta23(AT)gmail.com), Jan 16 2006

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000027, A049347, A057078, A079918.

Programs

Magma

I:=[3,2,1,6]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Sep 28 2017

Maple

seq(6*floor((n+2)/3)-n-2,n=1..72); # Dennis P. Walsh, Aug 16 2013

Mathematica

f[n_] := Switch[ Mod[n, 3], 0, n - 2, 1, n + 2, 2, n]; Array[f, 72] (* Robert G. Wilson v, Jan 18 2006 *)
LinearRecurrence[{1, 0, 1, -1}, {3, 2, 1, 6}, 100] (* or *) CoefficientList[Series[(3 - x - x^2 + 2 x^3) / ((1 + x + x^2) (1 - x)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 28 2017 *)
Reverse/@Partition[Range[81],3]//Flatten (* Harvey P. Dale, Oct 11 2020 *)

PARI

a(n)=2+n-2*((n+2)%3); \\ Jaume Oliver Lafont, Mar 25 2009

Formula

a(n) = 3*floor((n+2)/3) - (n-1) mod 3. - Robert G. Wilson v and Zak Seidov, Jan 20 2006

a(n) = a(n-3)+3 = a(n-1)+a(n-3)-a(n-4). - Jaume Oliver Lafont, Dec 02 2008

G.f.: (3*x - x^2 - x^3 + 2*x^4)/(1 - x - x^3 + x^4) = x*(3 - x - x^2 + 2*x^3)/((1 + x + x^2)*(1-x)^2). - Jaume Oliver Lafont, Mar 25 2009

a(n) = 6*floor((n+2)/3) - n - 2. - Dennis P. Walsh, Aug 16 2013

a(n) = A000027(n) + 2 * A057078(n+2). - Dennis P. Walsh, Aug 16 2013

a(n) = n + 2 * A079918(n-1) - 2 * A079918(n). - Dennis P. Walsh, Aug 16 2013

a(n) = n - 2*A049347(n). - Wesley Ivan Hurt, Sep 27 2017, simplified Jun 30 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - Amiram Eldar, Jan 31 2023

Extensions

More terms from Robert G. Wilson v, Jan 18 2006