A042964 -id:A042964 - cp's OEIS frontend

A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

2, 2, 6, 10, 18, 26, 38, 50, 66, 82, 102, 122, 146, 170, 198, 226, 258, 290, 326, 362, 402, 442, 486, 530, 578, 626, 678, 730, 786, 842, 902, 962, 1026, 1090, 1158, 1226, 1298, 1370, 1446, 1522, 1602, 1682, 1766, 1850, 1938, 2026, 2118

Offset: 0

Paul Curtz, Oct 14 2014

Numbers belonging to A016825: a(n) = A016825( A002620(n) ). - Bruno Berselli, Oct 15 2014

For n>1, a(n) is the number of row vectors of length 2n with entries in [1,n], first entry 1, with maximum inner distance. That is, vectors where the modular distance between adjacent entries is at least (n-2)/2. Modular distance is the minimum of remainders of (x - y) and (y - x) when dividing by n. Geometrically, this metric counts how far the integers mod n are from each other if 1 and n are adjacent as on a circle. - Omar Aceval Garcia, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000034, A000290, A002620, A016825, A042964, A080827, A168273, A005899, A069894.

[n^2+3/2+(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Oct 15 2014

Table[n^2 + 3/2 + (-1)^n/2, {n, 0, 50}] (* Bruno Berselli, Oct 15 2014 *)
CoefficientList[Series[2(x^3+x^2-x+1)/((1-x)^3 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2014 *)
LinearRecurrence[{2,0,-2,1},{2,2,6,10},60] (* Harvey P. Dale, Apr 08 2019 *)

Vec(-2*(x^3+x^2-x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014

[(2*n^2 +3 +(-1)^n)/2 for n in (0..50)] # G. C. Greubel, Dec 14 2021

a(n) = A000290(n) + A000034(n+1) = 4*A002620(n) + 2.
a(n+1) = 2*A080827(n+1) = (n+2)^2 - A042964(n+1) = a(n) + 2*n + 1 -(-1)^n.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Colin Barker, Oct 15 2014
G.f.: 2*(1-x+x^2+x^3) / ((1-x)^3*(x+1)). - Colin Barker, Oct 15 2014
E.g.f.: cosh(x) + (1 + x + x^2)*exp(x). - G. C. Greubel, Dec 14 2021
a(2n) = A005899(n) for n > 0, a(2n+1) = A069894(n). - Omar Aceval Garcia, Dec 30 2021

Typo in data fixed by Colin Barker, Oct 15 2014

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 4, 4, 5, 5, 6, 7, 6, 8, 9, 9, 11, 12, 13, 14, 15, 17, 19, 20, 23, 25, 27, 29, 31, 35, 37, 40, 46, 48, 52, 57, 60, 66, 71, 76, 85, 90, 97, 105, 112, 121, 129, 140, 152, 161, 174, 187, 198, 214, 228, 245, 265, 280, 302, 323, 342

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 2 or 3 mod 4.

			a(13) = 3 because we have [11, 2], [10, 3] and [7, 6].

Index entries for sequences related to partitions

Cf. A035366, A042964, A070048, A131795, A147599, A301504, A301505, A301507.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 2)) (1 + x^(4 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x^2, x^4] QPochhammer[-x^3, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042964(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A053438 Expansion of (1+x+2x^3)/((1-x)(1-x^2)).

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122

Offset: 0

N. J. A. Sloane, Jan 12 2000

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010684 (first differences), A263511 (partial sums).

I:=[2,3,6]; [1] cat [n le 3 select I[n] else Self(n-1) +Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, May 26 2018

A053438 := proc(n)
    if n > 0 then
        2*n -(1+(-1)^n)/2 ;
    else
        1 ;
    end if
end proc:
seq(A053438(n),n=0..30) ; # R. J. Mathar, Oct 27 2020

CoefficientList[Series[(1+x+2*x^3)/((1-x)*(1-x^2)), {x, 0, 50}], x] (* or *) Join[{1}, LinearRecurrence[{1,1,-1}, {2,3,6}, 50]] (* G. C. Greubel, May 26 2018 *)

a(n)=abs(n\2*4+n%2*3-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = 2*n -(1+(-1)^n)/2 if n>=1. - Frank Ellermann, Feb 12 2002
a(n) = A042964(n), n>0. - R. J. Mathar, Oct 13 2008
a(n) = A014601(n) - 1 for n>0. - Hugo Pfoertner, Oct 26 2020

A174091 a(n) = n OR 2.

Original entry on oeis.org

2, 3, 2, 3, 6, 7, 6, 7, 10, 11, 10, 11, 14, 15, 14, 15, 18, 19, 18, 19, 22, 23, 22, 23, 26, 27, 26, 27, 30, 31, 30, 31, 34, 35, 34, 35, 38, 39, 38, 39, 42, 43, 42, 43, 46, 47, 46, 47, 50, 51, 50, 51, 54, 55, 54, 55, 58, 59, 58, 59, 62, 63, 62, 63, 66, 67, 66

Offset: 0

Gary Detlefs, Feb 06 2013

OR(n, 2) + AND(n, 2) = n + 2.
OR(n, 2) - AND(n, 2) = n + 2*(-1)^floor(n/2), A004443.
a(n) = n when n = 2 or 3 mod 4 (n is in A042964). - Alonso del Arte, Feb 07 2013

			a(3) = 3 because OR(0011, 0010) = 0011 = 3.
a(4) = 6 because OR(0100, 0010) = 0110 = 6.
a(5) = 7 because OR(0101, 0010) = 0111 = 7.

Shane Chern, T. Cai, and H. Zhong, On the cardinality and sum of reciprocals of primitive sequences, Preprint 2018; To appear in Adv. Math. (China).
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. similar sequences listed in A244587.

Cf. A004443, A042964, A196521.

Maple
```
with(Bits): seq(Or(n,2), n=0..60);
```

Table[BitOr[n, 2], {n, 0, 100}] (* Alonso del Arte, Feb 06 2013 *)
LinearRecurrence[{2,-2,2,-1},{2,3,2,3},80] (* Harvey P. Dale, Oct 25 2016 *)

a(n)=bitor(n,2) \\ Charles R Greathouse IV, Feb 27 2013

a(n) = n + 1 + (-1)^floor(n/2).
G.f.: ( 2-x+x^3 ) / ( (1+x^2)*(x-1)^2 ). - R. J. Mathar, Feb 27 2013
Sum_{n>=0} (-1)^n/a(n) = Pi/4 - log(2)/2 = A196521. - Peter McNair, Aug 05 2023

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 5, 3, 9, 16, 6, 8, 10, 25, 15, 7, 11, 17, 26, 24, 14, 12, 18, 36, 49, 27, 23, 13, 19, 35, 37, 64, 48, 28, 22, 20, 34, 38, 50, 65, 63, 47, 29, 21, 33, 39, 51, 81, 100, 66, 62, 46, 30, 32, 40, 52, 80, 82, 121, 99, 67, 61, 45, 31, 41, 53, 79, 83, 101

Offset: 1

Boris Putievskiy, Mar 06 2013

Permutation of the natural numbers. a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.In general, let b(z) be a sequence of integer numbers. Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.

			The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
  ..........v...........v...........v
  >1....4...5..16..25..26..49..64..65...
  >2....3...6..15..24..27..48..63..66...
  .9....8...7..14..23..28..47..62..67...
  >10..11..12..13..22..29..46..61..68...
  >17..18..19..20..21..30..45..60..69...
  .36..35..34..33..32..31..44..59..70...
  >37..38..39..40..41..42..43..58..71...
  >50..51..52..53..54..55..56..57..72...
  .81..80..79..78..77..76..75..74..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  4,2;
  5,3,9;
  16,6,8,10;
  25,15,7,11,17;
  26,24,14,12,18,36;
  49,27,23,13,19,35,37;
  64,48,28,22,20,34,38,50;
  65,63,47,29,21,33,39,51,81;
  . . .

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j>=i:
   result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
else:
   result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2

For general case.
As table
T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
For this sequence b(z)=z^2 mod 3.
As table
T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, if  i >j;
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A214863 Numbers n such that n XOR 11 = n - 11.

Original entry on oeis.org

11, 15, 27, 31, 43, 47, 59, 63, 75, 79, 91, 95, 107, 111, 123, 127, 139, 143, 155, 159, 171, 175, 187, 191, 203, 207, 219, 223, 235, 239, 251, 255, 267, 271, 283, 287, 299, 303, 315, 319, 331, 335, 347, 351, 363

Offset: 1

Brad Clardy, Mar 09 2013

Links to sequences of the form n XOR m = n - m are found below with the value of m specified.

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

Cf. A005408 (m=1), A042964 (m=2), A131098 (m=3), A047566 (m=4), A047550 (m=5), A047589 (m=6), A004771 (m=7), A115419 (m=8), A214865 (m=9), A214864 (m=10), A133894 (m=12), A125169 (m=15).

Cf. also A016825, A168392.

XOR := func;
m:=11;
for n in [1 .. 500] do
      if (XOR(n, m) eq n-m) then n; end if;
end for;

Select[Range[400],BitXor[#,11]==#-11&] (* or *) LinearRecurrence[{1,1,-1},{11,15,27},50] (* Harvey P. Dale, Jun 05 2021 *)

a(n)= 1+8*n-2*(-1)^n.
a(n)=A016825(n) + A168392(n) +  for n>0.
G.f. x*(11+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 10 2013

A237989 Numbers m such that the numbers of primes, even positive integers and odd positive integers less than or equal to m are all odd.

Original entry on oeis.org

2, 6, 18, 26, 34, 42, 50, 70, 74, 78, 86, 98, 106, 110, 130, 138, 150, 158, 162, 170, 198, 214, 218, 222, 234, 238, 242, 246, 250, 258, 262, 270, 278, 286, 290, 310, 314, 334, 354, 358, 370, 382, 390, 394, 402, 406, 442, 450, 454, 462, 470, 474, 478, 490, 502

Offset: 1

Ivan N. Ianakiev, Feb 16 2014

			A cubic die whose faces are marked with the numbers from 1 to 6 has odd number of sides marked with prime numbers (2, 3 and 5), even integers (2, 4 and 6) and odd integers (1, 3 and 5). Therefore, 6 is in the sequence.

Ivan N. Ianakiev, Table of n, a(n) for n = 1..10000

Cf. A042963, A042964, A057812, A237990.

Select[Range[2, 1000, 4], OddQ[PrimePi[#]] &] (* Paolo Xausa, Jun 24 2024 *)

isok(n) = (primepi(n) % 2) && ((n % 4) == 2); \\ Michel Marcus, Mar 12 2014

Intersection of A042963 (odd number of odd numbers), A042964 (odd number of even numbers), A057812 (odd number of primes). - Michel Marcus, Feb 26 2014

A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Jan 18 2018

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Inverse: A292576.
Sequence of fixed points: A008586(n) for n > 0.
First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.
Subsequences:
  elements with odd index: A042963(A103889(n)) for n > 0.
  elements with even index A014601(n) for n > 0.
  odd elements: A166519(n-1) for n > 0.
  indices of odd elements: A042964(n) for n > 0.
  even elements: A005843(n) for n > 0.
  indices of even elements: A042948(n) for n > 0.
Other similar permutations: A116966, A284307, A292576.

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).
a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.
a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.
a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.
a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).
a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.
a(n) = A056699(n+1) - 1, n > 0.
a(n+2) = A168269(n+1) - a(n), n > 0.
a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
First differences: periodic, (1, -2, 3, 2) repeat.
Compositions:
  a(n) = A080412(A116966(n-1)) for n > 0.
  a(n) = A284307(A256008(n)) for n > 0.
  a(A067060(n)) = A133256(n) for n > 0.
  A116966(a(n+1)-1) = A092486(n) for n >= 0.
  A056699(a(n)) = A256008(n) for n > 0.

cp's OEIS Frontend

A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

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A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

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A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

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A301508 Expansion of Product_{k>=0} (1 + x^(4*k+2))*(1 + x^(4*k+3)).

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A053438 Expansion of (1+x+2*x^3)/((1-x)*(1-x^2)).

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A174091 a(n) = n OR 2.

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A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

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A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).

A053438 Expansion of (1+x+2x^3)/((1-x)(1-x^2)).