A004442 -id:A004442 - cp's OEIS frontend

A263824 Permutation of the nonnegative integers: [6k+3, 6k+4, 6k+5, 6k, 6k+1, 6k+2, ...].

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

Offset: 0

Wesley Ivan Hurt, Oct 27 2015

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A004442, A130151, A279318.

Magma
```
[n+3*(-1)^Floor(n/3) : n in [0..100]];
```

I:=[3,4,5,0,1]; [n le 5 select I[n] else 2*Self(n-1)- Self(n-2)-Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Nov 22 2015

A263824:=n->n+3*(-1)^floor(n/3): seq(A263824(n), n=0..100);

Table[n + 3 (-1)^Floor[n/3], {n, 0, 100}]
CoefficientList[Series[(3 - 2 x - 3 x^3 + 4 x^4)/((x - 1)^2 (1 + x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 22 2015 *)
LinearRecurrence[{2,-1,-1,2,-1},{3,4,5,0,1},70] (* Harvey P. Dale, Jun 23 2017 *)

Vec((3-2*x-3*x^3+4*x^4) / ((x-1)^2*(1+x^3)) + O(x^100)) \\ Altug Alkan, Oct 28 2015

A263824(n)=n+3*(-1)^(n\3) \\ M. F. Hasler, Nov 25 2015

G.f.: (3-2*x-3*x^3+4*x^4) / ((x-1)^2*(1+x^3)).
a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5), n>4.
a(n) = n + 3*(-1)^floor(n/3).
a(n) = a(n-6) + 6 for n>5. - Tom Edgar, Oct 28 2015
From Wesley Ivan Hurt, Nov 22 2015: (Start)
a(n) = n + 3*A130151(n).
a(3n) = 3*A004442(n). (End)
Sum_{n>=0, n!=3} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Dec 25 2023
E.g.f.: exp(-x) + x*exp(x) + 2*exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)). - Stefano Spezia, Aug 25 2025

A265376 a(1) = 1 and a(n) = Sum_{i=1..n-1} (-1)^iia(i).

Original entry on oeis.org

1, -1, -3, 6, 30, -120, -840, 5040, 45360, -362880, -3991680, 39916800, 518918400, -6227020800, -93405312000, 1307674368000, 22230464256000, -355687428096000, -6758061133824000, 121645100408832000, 2554547108585472000, -51090942171709440000, -1175091669949317120000

Offset: 1

José María Grau Ribas, Dec 07 2015

sign

1/abs(a(n)) + 1/abs(a(n+1)) = 1/(n-1)!, n = 3,5,7,... hence Sum_{n>1} 1/abs(a(n)) = cosh(1). - Peter McNair, Mar 04 2022

Cf. A004442, A001710 (b(n)=Sum_{i=1..n-1} i*b(i)).

a[1]=1; a[n_] := a[n] = Sum[(-1)^i*i*a[i], {i, 1, n - 1}]; Array[a,33]

For n>1, a(n) = (-1)^floor(n/2) * A001710(n) / floor(n/2). - Vaclav Kotesovec, Jan 26 2016

A307613 Inverse of the permutation A307485: one odd, two even, four odd, eight even, etc; extended with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 6, 9, 7, 10, 16, 11, 17, 12, 18, 13, 19, 14, 20, 15, 21, 32, 22, 33, 23, 34, 24, 35, 25, 36, 26, 37, 27, 38, 28, 39, 29, 40, 30, 41, 31, 42, 64, 43, 65, 44, 66, 45, 67, 46, 68, 47, 69, 48, 70, 49, 71, 50, 72, 51, 73, 52, 74, 53, 75, 54, 76, 55, 77

Offset: 0

M. F. Hasler, Apr 18 2019

nonn

See A307485 for further information, motivation & references.

Also, a(n) is the smallest k not yet in the sequence such that bitxor(k,a(n-1)) >= a(n-1). - Giorgos Kalogeropoulos, May 31 2019

			  Index n : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
A307485(n): 0, 1, 2, 4, 3, 5, 7, 9, 6, 8, ...
This sequence, the inverse permutation, is obtained by reading the above "from bottom to top", i.e., find the index in 2nd row, return the number above it: e.g., a(3) = 4, a(4) = 3, a(5) = 5, a(6) = 8, a(7) = 6, etc.

OEIS' Index to sequences related to permutations of the integers.

Cf. A307485 (inverse permutation), A307612 (partial sums thereof).
Cf. A103889 (odd & even swapped), A004442 (pairs reversed: n + (-1)^n).
Odd numbers: A005408. Even numbers: A005843.
Cf. A233275 (different permutation based on entangling odd & even numbers).

a[1]=1; a[n_] := a[n] = (t=1; While[BitXor[a[n-1],t] < a[n-1] || MemberQ[Array[a, n-1], t], t++]; t)
Join[{0}, Table[a[k], {k,100}]]  (* Giorgos Kalogeropoulos, May 31 2019 *)

my(A=apply(A307485,[1..99]), B=vecsort(A,,1)); for(i=1,#B,A[B[i]]==i||return(A307613=B[1..i-1]))

A354438 Square array A(n, k), n, k >= 0, read by antidiagonals; the factorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the factorial base expansions of n and k.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 4, 2, 4, 5, 5, 5, 5, 5, 5, 6, 4, 0, 4, 0, 4, 6, 7, 7, 1, 1, 1, 1, 7, 7, 8, 6, 8, 0, 2, 0, 8, 6, 8, 9, 9, 9, 9, 3, 3, 9, 9, 9, 9, 10, 8, 10, 8, 10, 2, 10, 8, 10, 8, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Rémy Sigrist, May 28 2022

The nonnegative integers together with A form an abelian group; A225901 gives inverse elements.

Each row is a permutation of the nonnegative integers.

			Square array A(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14
    2|   2   3   4   5   0   1   8   9  10  11   6   7  14  15  16  17
    3|   3   2   5   4   1   0   9   8  11  10   7   6  15  14  17  16
    4|   4   5   0   1   2   3  10  11   6   7   8   9  16  17  12  13
    5|   5   4   1   0   3   2  11  10   7   6   9   8  17  16  13  12
    6|   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21
    7|   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20
    8|   8   9  10  11   6   7  14  15  16  17  12  13  20  21  22  23
    9|   9   8  11  10   7   6  15  14  17  16  13  12  21  20  23  22
   10|  10  11   6   7   8   9  16  17  12  13  14  15  22  23  18  19
   11|  11  10   7   6   9   8  17  16  13  12  15  14  23  22  19  18
   12|  12  13  14  15  16  17  18  19  20  21  22  23   0   1   2   3
   13|  13  12  15  14  17  16  19  18  21  20  23  22   1   0   3   2
   14|  14  15  16  17  12  13  20  21  22  23  18  19   2   3   4   5
   15|  15  14  17  16  13  12  21  20  23  22  19  18   3   2   5   4

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Rémy Sigrist, Colored representation of the array A(n, k) for n, k < 7! (the hue is function of A(n, k), black pixels correspond to 0's)
Index entries for sequences related to factorial base representation

Cf. A003987, A004442, A108731, A225901, A354470 (primorial base analog).

A(n,k, s=i->i+1) = { my (v=0, f=1, r); for (i=1, oo, if (n==0 && k==0, return (v), r=s(i); v+=f*((n+k)%r); f*=r; n\=r; k\=r)) }

A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k).
A(n, 0) = n.
A(n, k) = 0 iff k = A225901(n).
A(n, 1) = A004442(n).

A354470 Square array A(n, k), n, k >= 0, read by antidiagonals; the primorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the primorial base expansions of n and k.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 4, 2, 4, 5, 5, 5, 5, 5, 5, 6, 4, 0, 4, 0, 4, 6, 7, 7, 1, 1, 1, 1, 7, 7, 8, 6, 8, 0, 2, 0, 8, 6, 8, 9, 9, 9, 9, 3, 3, 9, 9, 9, 9, 10, 8, 10, 8, 10, 2, 10, 8, 10, 8, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Rémy Sigrist, Jun 02 2022

The nonnegative integers together with A form an abelian group; A354469 gives inverse elements.

Each row is a permutation of the nonnegative integers.

			Square array A(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14
    2|   2   3   4   5   0   1   8   9  10  11   6   7  14  15  16  17
    3|   3   2   5   4   1   0   9   8  11  10   7   6  15  14  17  16
    4|   4   5   0   1   2   3  10  11   6   7   8   9  16  17  12  13
    5|   5   4   1   0   3   2  11  10   7   6   9   8  17  16  13  12
    6|   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21
    7|   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20
    8|   8   9  10  11   6   7  14  15  16  17  12  13  20  21  22  23
    9|   9   8  11  10   7   6  15  14  17  16  13  12  21  20  23  22
   10|  10  11   6   7   8   9  16  17  12  13  14  15  22  23  18  19
   11|  11  10   7   6   9   8  17  16  13  12  15  14  23  22  19  18
   12|  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27
   13|  13  12  15  14  17  16  19  18  21  20  23  22  25  24  27  26
   14|  14  15  16  17  12  13  20  21  22  23  18  19  26  27  28  29
   15|  15  14  17  16  13  12  21  20  23  22  19  18  27  26  29  28

Rémy Sigrist, Colored representation of the array A(n, k) for n, k < 2*3*5*7*11 (the hue is function of A(n, k), black pixels correspond to 0's)
Index entries for sequences related to primorial base

Cf. A004442, A235168, A354438 (factorial base analog), A354469.

A(n,k, s=i->prime(i)) = { my (v=0, f=1, r); for (i=1, oo, if (n==0 && k==0, return (v), r=s(i); v+=f*((n+k)%r); f*=r; n\=r; k\=r)) }

A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k).
A(n, 0) = n.
A(n, k) = 0 iff k = A354469(n).
A(n, 1) = A004442(n).

A004458 Nimsum n + 17.

Original entry on oeis.org

17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 81, 80, 83

Offset: 0

N. J. A. Sloane

A self-inverse permutation of the natural numbers. - Philippe Deléham, Nov 22 2016

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Nim-sums
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,-1).

Cf. A004442.

import Data.Bits (xor)
a004458 n = n `xor` 17 :: Integer  -- Reinhard Zumkeller, Nov 07 2012

Vec((17-x-14*x^2-15*x^16-x^17+18*x^18)/((1-x)^2*(1+x)*(1+x^16)) + O(x^50)) \\ Colin Barker, Apr 12 2016

a(n) = n + (-1)^n + 16*(-1)^(n\16); \\ Michel Marcus, Apr 12 2016

a(n) = n + (-1)^n + 16(-1)^[n/16]. - Mitchell Harris, Jan 10 2005
G.f.: (17-x-14*x^2-15*x^16-x^17+18*x^18) / ((1-x)^2*(1+x)*(1+x^16)). - Colin Barker, Apr 12 2016
a(n) = n XOR 17. - Michel Marcus, Apr 12 2016

A080418 Generalized Pascal triangle.

Original entry on oeis.org

1, 1, 3, 1, 2, 4, 1, 5, 5, 5, 1, 4, 11, 9, 6, 1, 7, 14, 21, 14, 7, 1, 6, 22, 34, 36, 20, 8, 1, 9, 27, 57, 69, 57, 27, 9, 1, 8, 37, 83, 127, 125, 85, 35, 10, 1, 11, 44, 121, 209, 253, 209, 121, 44, 11, 1, 10, 56, 164, 331, 461, 463, 329, 166, 54, 12

Offset: 1

Paul Barry, Feb 18 2003

			First rows are:
  {1},
  {1,3},
  {1,2,4},
  {1,5,5,5},
  {1,4,11,9,6},
  {1,7,14,21,14,7},
  ...
For example, 2 = 1 + 3 - 2, 5 = 1 + 2 + 2; 11 = 5 + 5 + 1, 14 = 4 + 11 - 1.

Joseph Adams, Table of n, a(n) for n = 1..4950

Columns include A000012, A004442, A000217+(-1)^n, A000292+(-1)^n and in general, binomial(n+k, k)+(-1)^n.

Diagonals include A000096, A063258.

t[n_, k_] := t[n, k]=Which[k==1, 1, n2, t[n-1, k-1] + t[n-1, k] + (-1)^(n+k)]; Flatten[Table[t[n, k], {n, 1, 20}, {k, 1, n}]] (* Frank M Jackson, Mar 27 2012 *)

T(n, 1)=1, T(n, k)=0 for k>n, T(n, 2) = T(n-1, 1) + T(n-1, 2) + 2*(-1)^n, T(n, k) = T(n-1, k-1) + T(n-1, k) + (-1)^(n+k) for k>2. [corrected by Frank M Jackson, Mar 27 2012]

Terms corrected and extended by Frank M Jackson, Mar 27 2012

A168195 a(n) = 2*n - a(n-1) + 1 with n>1, a(1)=5.

Original entry on oeis.org

5, 0, 7, 2, 9, 4, 11, 6, 13, 8, 15, 10, 17, 12, 19, 14, 21, 16, 23, 18, 25, 20, 27, 22, 29, 24, 31, 26, 33, 28, 35, 30, 37, 32, 39, 34, 41, 36, 43, 38, 45, 40, 47, 42, 49, 44, 51, 46, 53, 48, 55, 50, 57, 52, 59, 54, 61, 56, 63, 58, 65, 60, 67, 62, 69, 64, 71, 66, 73

Offset: 1

Vincenzo Librandi, Nov 20 2009

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

Cf. A004442, A147677.

I:=[5,0,7]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 27 2012

A168195:=n->n+1-3*(-1)^n; seq(A168195(n), n=1..100); # Wesley Ivan Hurt, Dec 13 2013

LinearRecurrence[{1,1,-1},{5,0,7},30] (* Vincenzo Librandi, Feb 27 2012 *)

From R. J. Mathar, Nov 22 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) = n + 1 - 3*(-1)^n.
G.f.: x*(5+2*x^2-5*x)/((1+x)*(x-1)^2). (End)
a(n) = n - 4 + 2^(2-(-1)^n). - Wesley Ivan Hurt, Dec 13 2013
a(n) = A004442(n-1) + 2*(1-(-1)^n) = A147677(n) - floor((n+3)/2). - Filip Zaludek, Oct 31 2016
Sum_{n>=3} (-1)^n/a(n) = 23/15 - log(2). - Amiram Eldar, Feb 23 2023

A179783 a(n) = 2n(n+1)*(n+2)/3 + (-1)^n.

Original entry on oeis.org

1, 3, 17, 39, 81, 139, 225, 335, 481, 659, 881, 1143, 1457, 1819, 2241, 2719, 3265, 3875, 4561, 5319, 6161, 7083, 8097, 9199, 10401, 11699, 13105, 14615, 16241, 17979, 19841, 21823, 23937, 26179, 28561, 31079

Offset: 0

Bruno Berselli, Jul 29 2010 - Sep 07 2010

First differences in 2*A081352.

Second differences in 4*A004442.

B. Berselli, Table of n, a(n) for n = 0..10000.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A005744, A026035, A175109, A131941.

[(2/3)*n*(n+1)*(n+2)+(-1)^n: n in [0..35]];

LinearRecurrence[{3,-2,-2,3,-1},{1,3,17,39,81},40] (* Harvey P. Dale, Mar 04 2023 *)

for(n=0, 35, print1((2/3)*n*(n+1)*(n+2)+(-1)^n", "));

G.f.: (1+10*x^2-4*x^3+x^4)/((1+x)*(1-x)^4); exp(-x)+(2/3)*exp(x)*x*(6+6*x+x^2).
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.
a(n) = 4*A000292(n)+(-1)^n.

A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

Original entry on oeis.org

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

Offset: 0

Wesley Ivan Hurt, Oct 17 2015

Fixed points are the odd numbers (A005408).

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.

[n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];

/* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // Bruno Berselli, Nov 08 2015

A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);

Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]

Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

G.f.: (2 - 3*x + 2*x^2 + x^3)/((x - 1)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
a(n) = n + (1 + (-1)^n)*(-1)^(n*(n+1)/2).
a(n) = 4*floor((n+1)/4) - (n mod 4) + 2.
a(n) = A092486(n) - 1.
a(n) = n + A176742(n) for n>0.
a(2n) = 2*A004442(n), a(2n+1) = A005408(n).
a(-n-1) = -A263449(n).
a(n+1) = a(n) - A132429(n+1)*(-1)^n.
Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Dec 25 2023

cp's OEIS Frontend

A263824 Permutation of the nonnegative integers: [6k+3, 6k+4, 6k+5, 6k, 6k+1, 6k+2, ...].

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A265376 a(1) = 1 and a(n) = Sum_{i=1..n-1} (-1)^i*i*a(i).

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A307613 Inverse of the permutation A307485: one odd, two even, four odd, eight even, etc; extended with a(0) = 0.

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A354438 Square array A(n, k), n, k >= 0, read by antidiagonals; the factorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the factorial base expansions of n and k.

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A354470 Square array A(n, k), n, k >= 0, read by antidiagonals; the primorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the primorial base expansions of n and k.

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A004458 Nimsum n + 17.

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A080418 Generalized Pascal triangle.

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A265376 a(1) = 1 and a(n) = Sum_{i=1..n-1} (-1)^iia(i).

A179783 a(n) = 2n(n+1)*(n+2)/3 + (-1)^n.