A083220 -id:A083220 - 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

A083219 a(n) = n - 2*floor(n/4).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Reinhard Zumkeller, Apr 22 2003

Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - Michel Lagneau, Apr 09 2013

a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - Kevin Ryde, Dec 13 2019

Altug Alkan, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A083220, A129756, A162751 (second highest bit removed).
Cf. A021913, A162330, A285869.
Essentially the same as A018837.

a:= n-> n - 2*floor(n/4):
seq(a(n), n=0..74);  # Alois P. Heinz, Jan 24 2021

Array[# - 2 Floor[#/4] &, 75, 0] (* Michael De Vlieger, Oct 02 2017 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* Harvey P. Dale, Oct 01 2018 *)

a(n) = n - n\4*2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A083220(n)/2.
a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.
G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Aug 28 2008
a(n) = n - A129756(n). - Michel Lagneau, Apr 09 2013
Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - Wolfdieter Lang, May 08 2017
a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - Kevin Ryde, Dec 13 2019
E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - Stefano Spezia, May 27 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - Amiram Eldar, Aug 21 2023

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

A267573 a(n) = prime(n) + (prime(n) mod 4).

Original entry on oeis.org

4, 6, 6, 10, 14, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 62, 62, 70, 74, 74, 82, 86, 90, 98, 102, 106, 110, 110, 114, 130, 134, 138, 142, 150, 154, 158, 166, 170, 174, 182, 182, 194, 194, 198, 202, 214, 226, 230, 230, 234, 242, 242, 254, 258, 266, 270

Offset: 1

Emre APARI, Jan 17 2016

The primes corresponding to the cases where a(n) = a(n+1) can be found in A071698. - Michel Marcus, Jan 17 2016

			p=19; 19 + (19 modulo 4) = 22.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A039702, A071698, A083220.

[NthPrime(n)+(NthPrime(n) mod 4): n in [1..100]]; // Vincenzo Librandi, Jan 17 2016

A267573:=n->ithprime(n)+(ithprime(n) mod 4): seq(A267573(n), n=1..100); # Wesley Ivan Hurt, Jan 17 2016

Table[Prime[n] + Mod[Prime[n], 4], {n, 60}] (* Vincenzo Librandi, Jan 17 2016 *)
#+Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Jun 12 2020 *)

a(n) = prime(n) + (prime(n) % 4); \\ Michel Marcus, Jan 17 2016

lista(nn) = forprime(p=2, nn, print1(p + p % 4, ", ")); \\ Altug Alkan, Jan 17 2016

a(n) = A000040(n) + A039702(n).

a(n) = A083220(prime(n)). - Michel Marcus, Jan 17 2016

More terms from Vincenzo Librandi, Jan 17 2016

A282848 a(n) = 2*n + 1 + n mod 4.

Original entry on oeis.org

4, 7, 10, 9, 12, 15, 18, 17, 20, 23, 26, 25, 28, 31, 34, 33, 36, 39, 42, 41, 44, 47, 50, 49, 52, 55, 58, 57, 60, 63, 66, 65, 68, 71, 74, 73, 76, 79, 82, 81, 84, 87, 90, 89, 92, 95, 98, 97, 100, 103, 106, 105, 108, 111, 114, 113, 116, 119, 122, 121, 124, 127

Offset: 1

Zak Seidov, Feb 23 2017

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

Cf. A143097, A083220.

Table[2*k + 1 + Mod[k, 4], {k, 100}]
CoefficientList[ Series[(4 + 3 x + 3 x^2 - x^3 - x^4)/((x -1)^2 (1 + x + x^2 + x^3)), {x, 0, 60}], x] (* or *)
LinearRecurrence[{1, 0, 0, 1, -1}, {4, 7, 10, 9, 12}, 70] (* Robert G. Wilson v, Feb 23 2017 *)

Vec(x*(4 + 3*x + 3*x^2 - x^3 - x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^100)) \\ Colin Barker, Feb 23 2017

For n > 4 a(n) = a(n - 4) + 8.

G.f.: x*(4 + 3*x + 3*x^2 - x^3 - x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)). - Colin Barker, Feb 23 2017

cp's OEIS Frontend

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

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A083219 a(n) = n - 2*floor(n/4).

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A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

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A267573 a(n) = prime(n) + (prime(n) mod 4).

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A282848 a(n) = 2*n + 1 + n mod 4.

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

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.