A115302 -id:A115302 - cp's OEIS frontend

A143097 3k - 2 interleaved with 3k - 1 and 3*k.

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

Offset: 1

Gary W. Adamson, Jul 24 2008

First differences give A143098.

Binomial transform = A143099: (1, 3, 9, 22, 50, 113, 256, ...).

			Interleave 3 subsets:
  1,....4,.......7,......10,......13,......16,...
  ...2,.......5,.......8,......11,......14,...
  .........3,.......6,.......9,......12,...
  ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A115302, A143098, A143099.

Cf. A083220 (n + (n mod 4)). - Zak Seidov, Feb 23 2017

A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: seq(A143097(n), n=1..70); # Nathaniel Johnston, Apr 30 2011

With[{nn=70},Join[{1},Riffle[Rest[Select[Range[nn],!Divisible[#,3]&]], Range[ 3,nn,3],3]]] (* Harvey P. Dale, May 06 2012 *)
Table[If[k == 1, 1, k - 1 + Mod[k - 1, 3]], {k, 100}] (* Zak Seidov, Feb 23 2017 *)

A permutation of the natural numbers: 3*k - 2 interleaved with 3*k - 1 and 3*k; k=1,2,3,...; given a(1) = 1. a(n) = n if the subset = 3*k - 1: (2, 5, 8, ...); a(n) = n+1 in 3*k - 2, k>1: (4, 7, 10, ...); and a(n) = (n-1) in 3*k: (3, 6, 9, ...).
G.f.: x(1+x+2x^2-2x^3+x^4)/((1-x)^2(1+x+x^2)). - R. J. Mathar, Sep 06 2008
a(n) = if(n==1, 1, (n-1) + (n-1) mod 3). - Zak Seidov, Feb 23 2017
For n>1, a(n) = n+2*sin(2*(n+1)*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Sep 27 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - 2*Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Aug 21 2023

A116080 Permutation of natural numbers generated by 4-rowed array shown below.

Original entry on oeis.org

0, 4, 1, 8, 5, 2, 12, 9, 6, 3, 16, 13, 10, 7, 20, 17, 14, 11, 24, 21, 18, 15, 28, 25, 22, 19, 32, 29, 26, 23, 36, 33, 30, 27, 40, 37, 34, 31, 44, 41, 38, 35, 48, 45, 42, 39, 52, 49, 46, 43, 56, 53, 50, 47, 60, 57, 54, 51, 64, 61, 58, 55, 68, 65, 62, 59, 72, 69, 66, 63, 76, 73, 70

Offset: 1

Giovanni Teofilatto, Mar 12 2006

4  8 12 16 20 24 28 32 ... a(n) = 4n   => A008586;
5  9 13 17 21 25 29 33 ... a(n) = 4n+1 => A016813;
6 10 14 18 22 26 30 34 ... a(n) = 4n+2 => A016825;
7 11 15 19 23 27 31 35 ... a(n) = 4n+3 => A004767.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1).

Cf. A115302.

LinearRecurrence[{1,0,0,1,-1},{0,4,1,8,5,2,12,9,6,3,16},80] (* Harvey P. Dale, Feb 20 2022 *)

For n > 0, a(n+5) = a(n) + 8 iff a(n+5)= 1.
a(n)= a(n-1) + a(n-4) - a(n-5), n>=12. - R. J. Mathar, Apr 22 2010
G.f.: x^2*(4-3*x+7*x^2-3*x^3-7*x^4+13*x^5-10*x^6+3*x^9)/(1-x-x^4+x^5). - Philippe Deléham, Dec 02 2016

Corrected (47 replaced by 41) by R. J. Mathar, Apr 22 2010

A115731 Permutation of natural numbers generated by 3-rowed array shown below.

Original entry on oeis.org

1, 6, 2, 7, 5, 3, 12, 8, 4, 13, 11, 9, 18, 14, 10, 19, 17, 15, 24, 20, 16, 25, 23, 21, 30, 26, 22, 31, 29, 27, 36, 32, 28, 37, 35, 33, 42, 38, 34, 43, 41, 39, 48, 44, 40, 49, 47, 45, 54, 50, 46, 55, 53, 51, 60, 56, 52, 61, 59, 57, 66, 62, 58, 67, 65, 63, 72, 68, 64, 73, 71, 69

Offset: 1

Giovanni Teofilatto, Mar 19 2006

6 7 12 13 18 19 24 25 ... a(n)=congruent to {0, 1} mod 6, Cf. A047225.
5 8 11 14 17 20 23 26 ... a(n)= 3n+2= A016789.
4 9 10 15 16 21 22 27 ... a(n)=congruent to {3, 4} mod 6, Cf. A047230.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A115302.

Vec(x*(1 + 4*x - 8*x^2 + 13*x^3 - 15*x^4 + 13*x^5 - 5*x^6 - 4*x^7 + 4*x^8) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)) + O(x^80)) \\ Colin Barker, Apr 01 2018

Starting with the term a(3), a(n+6k) = a(n) + 6k, with k>=1.
From Colin Barker, Apr 01 2018: (Start)
G.f.: x*(1 + 4*x - 8*x^2 + 13*x^3 - 15*x^4 + 13*x^5 - 5*x^6 - 4*x^7 + 4*x^8) / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>9.
(End)

A116547 Permutation of natural numbers generated by 5-rowed array shown below.

Original entry on oeis.org

0, 5, 1, 10, 6, 2, 15, 11, 7, 3, 20, 16, 12, 8, 4, 25, 21, 17, 13, 9, 30, 26, 22, 18, 14, 35, 31, 27, 23, 19, 40, 36, 32, 28, 24, 45, 41, 37, 33, 29, 50, 46, 42, 38, 34, 55, 51, 47, 43, 39, 60, 56, 52, 48, 44, 65, 61, 57, 53, 49, 70, 66, 62, 58, 54, 75, 71, 67, 63, 59, 80, 76, 72

Offset: 1

Giovanni Teofilatto, Mar 16 2006

nonn

5 10 15 20 25 30 35 40 45 ... a(n)=5n
6 11 16 21 26 31 36 41 46 ... a(n)=5n+1
7 12 17 22 27 32 37 42 47 ... a(n)=5n+2
8 13 18 23 28 33 38 43 48 ... a(n)=5n+3
9 14 19 24 29 34 39 44 49 ... a(n)=5n+4

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A115302, A116080.

0, 5, 1, 10, 6, 2, 15, 11, 7, 3, seq(seq(t + 5*k,t=[15,11,7,3,-1]),k=1..20); # Robert Israel, Mar 23 2017

Select[Join[{0},Table[Most[NestList[#-4&,5n,5]],{n,20}]]//Flatten,#>=0&] (* Harvey P. Dale, Nov 26 2016 *)

Starting with term a(10): a(n+5k) = a(n) + 5k, with k>=1.

G.f.: x*(5-4*x+9*x^2-4*x^3-4*x^4+8*x^5-13*x^7+21*x^9-17*x^10+4*x^14)/(1-x-x^5+x^6). - Robert Israel, Mar 23 2017

A116551 Permutation of natural numbers generated by 3-rowed array shown below.

Original entry on oeis.org

0, 3, 1, 6, 4, 2, 9, 7, 5, 12, 10, 8, 15, 13, 11, 18, 16, 14, 21, 19, 17, 24, 22, 20, 27, 25, 23, 30, 28, 26, 33, 31, 29, 36, 34, 32, 39, 37, 35, 42, 40, 38, 45, 43, 41, 48, 46, 44, 51, 49, 47, 54, 52, 50, 57, 55, 53, 60, 58, 56, 63, 61, 59, 66, 64, 62, 69, 67, 65, 72, 70, 68, 75

Offset: 1

Giovanni Teofilatto, Mar 17 2006

3 6 9 12 15 18 21 24 ... a(n)= 3n
4 7 10 13 16 19 22 25 ... a(n)= 3n+1
5 8 11 14 17 20 23 26 ... a(n)= 3n+2

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A115302.

Rest[CoefficientList[Series[x^2*(2*x^5 - 5*x^3 + 5*x^2 - 2*x + 3)/(x^4 - x^3 - x + 1), {x,0,50}], x]] (* or *) Join[{0, 3, 1, 6, 4, 2, 9}, LinearRecurrence[{1,0,1,-1}, {7, 5, 12, 10}, 50]] (* G. C. Greubel, Sep 20 2017 *)

x='x+O('x^50); Vec(x^2*(2*x^5 - 5*x^3 + 5*x^2 - 2*x + 3)/(x^4 - x^3 - x + 1)) \\ G. C. Greubel, Sep 20 2017

Starting at the term a(3), a(n+3k) = a(n) + 3k, with k>=1.
From Chai Wah Wu, Jul 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 7.
G.f.: x^2*(2*x^5 - 5*x^3 + 5*x^2 - 2*x + 3)/(x^4 - x^3 - x + 1). (End)

A112303 Permutation of primes generated by 3-rowed array shown below.

Original entry on oeis.org

2, 7, 3, 17, 11, 5, 29, 19, 13, 41, 31, 23, 53, 43, 37, 67, 59, 47, 79, 71, 61, 97, 83, 73, 107, 101, 89, 127, 109, 103, 139, 139, 131, 113, 157, 149, 137, 173, 163, 151, 191, 179, 167, 193, 181, 227, 211, 197, 239, 229, 223, 257, 263, 251

Offset: 1

Giovanni Teofilatto, Oct 17 2006, Nov 08 2006

7 17 29 41 53 67 79...(A031377)
11 19 31 43 59 71 83...(A031369)
13 23 37 47 61 73 89...(A031336)

Cf. A115302.

a(n) = A000040(a(p+3q) = a(p)+3q) with p and q positive integers.

a(n) = PrimePi(a(p+3q) = a(p)+3q) with p and q positive integers.

cp's OEIS Frontend

A143097 3*k - 2 interleaved with 3*k - 1 and 3*k.

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A116080 Permutation of natural numbers generated by 4-rowed array shown below.

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A115731 Permutation of natural numbers generated by 3-rowed array shown below.

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A116547 Permutation of natural numbers generated by 5-rowed array shown below.

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A116551 Permutation of natural numbers generated by 3-rowed array shown below.

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A112303 Permutation of primes generated by 3-rowed array shown below.

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A143097 3k - 2 interleaved with 3k - 1 and 3*k.