A006369 -id:A006369 - cp's OEIS frontend

A168223 a(n) = A006369(n) - A006368(n).

0, 0, 0, 0, -1, 3, -5, 4, -1, -1, -2, 7, -10, 7, -2, -1, -3, 10, -15, 11, -3, -2, -4, 14, -20, 14, -4, -2, -5, 17, -25, 18, -5, -3, -6, 21, -30, 21, -6, -3, -7, 24, -35, 25, -7, -4, -8, 28, -40, 28, -8, -4, -9, 31, -45, 32, -9, -5, -10, 35, -50, 35, -10, -5, -11, 38, -55, 39

Offset: 0

Reinhard Zumkeller, Nov 20 2009

A047342 and A168223 give record values and where they occur: a(A168224(n))=A047342(n) and a(m) < A047342(n) for m < A168224(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (-2,-2,0,3,4,3,0,-2,-2,-1).

a168223 n = a006369 n - a006368 n  -- Reinhard Zumkeller, Mar 15 2014

LinearRecurrence[{-2,-2,0,3,4,3,0,-2,-2,-1},{0, 0, 0, 0, -1, 3, -5, 4, -1, -1},50] (* G. C. Greubel, Jul 16 2016 *)

a(12*n) = -10*n,          a(12*n+1) = 7*n.
a(12*n+2) = -2*n,         a(12*n+3) = -n.
a(12*n+4) = -2*n - 1,     a(12*n+5) = 7*n + 3.
a(12*n+6) = -10*n - 5,    a(12*n+7) = 7*n + 4.
a(12*n+8) = -2*n -1,      a(12*n+9) = -n - 1.
a(12*n+10) = -2*n - 2,    a(12*n+11) = 7*n + 7.
G.f.: -x^4*(x^2-x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^2+x+1)^2). - Colin Barker, Apr 04 2013

A349368 Dirichlet inverse of A006369, the inverse permutation of "amusical permutation".

Original entry on oeis.org

1, -3, -2, 4, -7, 8, -9, -8, -2, 29, -15, -18, -17, 35, 18, 16, -23, 4, -25, -68, 22, 61, -31, 44, 16, 67, -2, -76, -39, -104, -41, -32, 38, 93, 79, 6, -49, 99, 42, 168, -55, -120, -57, -140, 10, 125, -63, -104, 16, -128, 58, -148, -71, 0, 137, 184, 62, 157, -79, 354, -81, 163, 14, 64, 151, -216, -89, -212, 78, -445

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006369, A349369.

Cf. also A349351, A349376.

A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A006369(n/d) * a(d).

a(n) = A349369(n) - A006369(n).

A349369 Sum of A006369 and its Dirichlet inverse, where A006369 is the inverse of "amusical permutation", A006368.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 3, 4, 42, 0, -10, 0, 54, 28, 37, 0, 16, 0, -41, 36, 90, 0, 60, 49, 102, 16, -39, 0, -84, 0, 11, 60, 138, 126, 30, 0, 150, 68, 221, 0, -92, 0, -81, 40, 186, 0, -72, 81, -61, 92, -79, 0, 36, 210, 259, 100, 234, 0, 394, 0, 246, 56, 149, 238, -172, 0, -121, 124, -352, 0, 8, 0, 294, -22, -119, 270

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006369, A349368.

Cf. also A349352, A349378.

A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));
A349369(n) = (A006369(n)+A349368(n));

a(n) = A006369(n) + A349368(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A006369(d) * A349368(n/d).

A223084 Trajectory of 80 under the map n-> A006369(n).

Original entry on oeis.org

80, 107, 143, 191, 255, 170, 227, 303, 202, 269, 359, 479, 639, 426, 284, 379, 505, 673, 897, 598, 797, 1063, 1417, 1889, 2519, 3359, 4479, 2986, 3981, 2654, 3539, 4719, 3146, 4195, 5593, 7457, 9943, 13257, 8838, 5892, 3928, 5237, 6983, 9311, 12415, 16553, 22071

Offset: 1

N. J. A. Sloane, Mar 22 2013

nonn

It is conjectured that this trajectory does not close on itself.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A006369, A006368, A182205.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[80];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {80}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {80}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A223085 Trajectory of 82 under the map n-> A006369(n).

Original entry on oeis.org

82, 109, 145, 193, 257, 343, 457, 609, 406, 541, 721, 961, 1281, 854, 1139, 1519, 2025, 1350, 900, 600, 400, 533, 711, 474, 316, 421, 561, 374, 499, 665, 887, 1183, 1577, 2103, 1402, 1869, 1246, 1661, 2215, 2953, 3937, 5249, 6999, 4666, 6221, 8295, 5530, 7373

Offset: 1

N. J. A. Sloane, Mar 22 2013

nonn

It is conjectured that this trajectory does not close on itself.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A006369, A006368, A182205.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[82];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {82}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {82}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)
NestList[If[Divisible[#,3],(2#)/3,Floor[(4#)/3+1/2]]&,82,50] (* Harvey P. Dale, Sep 22 2019 *)

A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 5, 3, 3, 1, 0, 6, 7, 7, 2, 2, 1, 0, 7, 4, 9, 9, 3, 3, 1, 0, 8, 9, 5, 6, 6, 2, 2, 1, 0, 9, 11, 6, 7, 4, 4, 3, 3, 1, 0, 10, 6, 15, 4, 9, 5, 5, 2, 2, 1, 0, 11, 13, 4, 10, 5, 6, 7, 7, 3, 3, 1, 0, 12, 15, 17, 5, 13, 7, 4, 9, 9, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 18 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,   0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,   3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,   2,  3, ... = A176059
  [ 4]   4,  5,  7,  9,  6,  4,  5,  7,  9,   6,  4, ... = A094328
  [ 5]   5,  7,  9,  6,  4,  5,  7,  9,  6,   4,  5, ... = A094328 (shifted)
  [ 6]   6,  4,  5,  7,  9,  6,  4,  5,  7,   9,  6, ... = A094328 (shifted)
  [ 7]   7,  9,  6,  4,  5,  7,  9,  6,  4,   5,  7, ... = A094328 (shifted)
  [ 8]   8, 11, 15, 10, 13, 17, 23, 31, 41,  55, 73, ... = A028394
  [ 9]   9,  6,  4,  5,  7,  9,  6,  4,  5,   7,  9, ... = A094328 (shifted)
  [10]  10, 13, 17, 23, 31, 41, 55, 73, 97, 129, 86, ... = A028394 (shifted)
  ...    |   |   |
      A001477|A168222
          A006369

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693, A028394, A028396, A094328, A094329, A176059, A185589, A185590, A217729, A223083, A223084, A223085.
Columns: A001477, A006369, A168222.
Main diagonal: A368228.
Cf. A368179.

A006369[n_]:=If[Divisible[n,3],2n/3,Round[4n/3]];
A368227list[dmax_]:=With[{a=Reverse[Table[NestList[A006369,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368227list[15] (* Generates 15 antidiagonals *)

A368228 Main diagonal of A368227: the n-th term in the trajectory of n under the A006369 map.

Original entry on oeis.org

0, 1, 2, 2, 6, 5, 4, 6, 41, 7, 86, 129, 97, 68, 49, 68, 49, 287, 102, 137, 102, 137, 385, 538, 183, 513, 361, 511, 338, 481, 681, 1791, 161, 855, 605, 801, 271, 751, 538, 356, 1939, 1325, 890, 637, 111, 1194, 380, 2111, 755, 1977, 163, 1887, 601, 1701, 563, 12403, 513, 1491, 7802

Offset: 0

Paolo Xausa, Dec 18 2023

Cf. A006369, A368179, A368180, A368227.

A368228[n_]:=Nest[If[Divisible[#,3],2#/3,Round[4#/3]]&,n,n];
Array[A368228,100,0]

a(n) = A368227(n,n).

A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 9, 5, 12, 7, 15, 8, 18, 10, 21, 11, 24, 13, 27, 14, 30, 16, 33, 17, 36, 19, 39, 20, 42, 22, 45, 23, 48, 25, 51, 26, 54, 28, 57, 29, 60, 31, 63, 32, 66, 34, 69, 35, 72, 37, 75, 38, 78, 40, 81, 41, 84, 43, 87, 44, 90, 46, 93, 47, 96, 49, 99, 50, 102, 52, 105, 53

Offset: 0

N. J. A. Sloane and J. H. Conway

A permutation of the nonnegative integers.
There is a famous open question concerning the closed trajectories under this map - see A217218, A028393, A028394, and Conway (2013).
This is lodumo_3 of A131743. - Philippe Deléham, Oct 24 2011
Multiples of 3 interspersed with numbers other than multiples of 3. - Harvey P. Dale, Dec 16 2011
For n>0: a(2n+1) is the smallest number missing from {a(0),...,a(2n-1)} and a(2n) = a(2n-1) + a(2n+1). - Bob Selcoe, May 24 2017
From Wolfdieter Lang, Sep 21 2021: (Start)
The permutation P of positive natural numbers with P(n) = a(n-1) + 1, for n >= 1, is the inverse of the permutation given in A265667, and it maps the index n of A178414 to the index of A047529: A178414(n) = A047529(P(n)).
Thus each number {1, 3, 7} (mod 8) appears in the first column A178414 of the array A178415 just once. For the formulas see below. (End)
Starting at n = 1, the sequence equals the smallest unused positive number such that a(n)-a(n-1) does not appear as a term in the current sequence. - Scott R. Shannon, Dec 20 2023

			9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.

J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, CO, 1972, pp. 49-52.
R. K. Guy, Unsolved Problems in Number Theory, E17.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Zumkeller, Table of n, a(n) for n = 0..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7..
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198. [Introduces the name "amusical permutation".]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
S. Schreiber & N. J. A. Sloane, Correspondence, 1980
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for two-way infinite sequences
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Inverse mapping to A006369.
Cf. A047529, A178414, A178415, A265667.
Cf. A028393, A028294, A028397, A180853, A180864, A182205, A217218.
Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

a006368 n | u' == 0   = 3 * u
          | otherwise = 3 * v + (v' + 1) `div` 2
          where (u,u') = divMod n 2; (v,v') = divMod n 4
-- Reinhard Zumkeller, Apr 18 2012

[n mod 2 eq 1 select Round(3*n/4) else 3*n/2: n in [0..80]]; // G. C. Greubel, Jan 03 2024

f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi; # N. J. A. Sloane, Jan 21 2011
A006368:=(1+3*z+z**2+3*z**3+z**4)/(1+z**2)/(z-1)**2/(1+z)**2; # [Conjectured (correctly, except for the offset) by Simon Plouffe in his 1992 dissertation.]

Table[If[EvenQ[n],(3n)/2,Floor[(3n+2)/4]],{n,0,80}] (* or *) LinearRecurrence[ {0,1,0,1,0,-1},{0,1,3,2,6,4},80] (* Harvey P. Dale, Dec 16 2011 *)

PARI
```
a(n)=(3*n+n%2)\(2+n%2*2)
```
PARI
```
a(n)=if(n%2,round(3*n/4),3*n/2)
```

def a(n): return 0 if n == 0 else 3*n//2 if n%2 == 0 else (3*n+1)//4
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 12 2021

If n even, then a(n) = 3*n/2, otherwise, a(n) = round(3*n/4).
G.f.: x*(1+3*x+x^2+3*x^3+x^4)/((1-x^2)*(1-x^4)). - Michael Somos, Jul 23 2002
a(n) = -a(-n).
From Reinhard Zumkeller, Nov 20 2009: (Start)
a(n) = A006369(n) - A168223(n).
A168221(n) = a(a(n)).
A168222(a(n)) = A006369(n). (End)
a(n) = a(n-2) + a(n-4) - a(n-6); a(0)=0, a(1)=1, a(2)=3, a(3)=2, a(4)=6, a(5)=4. - Harvey P. Dale, Dec 16 2011
From Wolfdieter Lang, Sep 21 2021: (Start)
Formulas for the permutation P(n) = a(n-1) + 1 mentioned above:
P(n) = n + floor(n/2) if n is odd, and n - floor(n/4) if n is even.
P(n) = (3*n-1)/2 if n is odd; P(n) = (3*n+2)/4 if n == 2 (mod 4); and P(n) = 3*n/4 if n == 0 (mod 4). (End)

Edited by Michael Somos, Jul 23 2002

I replaced the definition with the original definition of Conway and Guy. - N. J. A. Sloane, Oct 03 2012

A004396 One even number followed by two odd numbers.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47

Offset: 0

N. J. A. Sloane

Maximal number of points on a triangular grid of edge length n-1 with no 2 points on same row, column, or diagonal. See Problem 252 in The Inquisitive Problem Solver. - R. K. Guy [Comment revised by N. J. A. Sloane, Jul 01 2016]
See also Problem C2 of 2009 International Mathematical Olympiad. - Ruediger Jehn, Oct 19 2021
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(3).
Starting at 3, 3, ..., gives maximal number of acute angles in an n-gon. - Takenov Nurdin (takenov_vert(AT)e-mail.ru), Mar 04 2003
Let b(1) = b(2) = 1, b(k) = b(k-1)+( b(k-2) reduced (mod 2)); then a(n) = b(n-1). - Benoit Cloitre, Aug 14 2002
(1+x+x^2+x^3 ) / ( (1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for Sigma_4.
For n > 6, maximum number of knight moves to reach any square from the corner of an (n-2) X (n-2) chessboard. Likewise for n > 6, the maximum number of knight moves to reach any square from the middle of an (2n-5) X (2n-5) chessboard. - Ralf Stephan, Sep 15 2004
A transform of the Jacobsthal numbers A001045 under the mapping of g.f.s g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
For n >= 1; a(n) = number of successive terms of A040001 that add to n; or length of n-th term of A028359. - Jaroslav Krizek, Mar 28 2010
For n > 0: a(n) = length of n-th row in A082870. - Reinhard Zumkeller, Apr 13 2014
Also the independence number of the n-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
In a game of basketball points can be accumulated by making field goals (two or three points) or free throws (one point). a(n) is the number of different ways to score n-1 points. For example, a score of 4 can be achieved in 3 different ways, with 2 shots (3+1 or 2+2), 3 shots (2+1+1) or 4 shots (1+1+1+1), so a(5) = 3. - Ivan N. Ianakiev, Mar 31 2025

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + ...

J. Kurschak, Hungarian Mathematical Olympiads, 1976, Mir, Moscow.
Paul Vanderlind, Richard K. Guy, and Loren C. Larson, The Inquisitive Problem Solver, MAA, 2002. See Problem 252.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 246.
Art of Problem Solving Forum, Ordered triples choosing. - _Joel B. Lewis_, May 21 2009
J. Choi and N. Pippenger, Counting the Angels and Devils in Escher's Circle Limit IV, arXiv preprint arXiv:1310.1357 [math.CO], 2013.
C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes, Discrete Math., 9 (1974), 391-400 (see proof of Theorem 1).
Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See Eq. (4).
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Independence Number.
50th International Mathematical Olympiad 2009, Problem Shortlist with Solutions.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index to sequences related to Olympiads.

Cf. A001045, A002620, A004523, A004773, A006369, A008620, A032766, A040001, A082870, A092200, A096777.

a004396 n = a004396_list !! n
a004396_list = 0 : 1 : 1 : map (+ 2) a004396_list
-- Reinhard Zumkeller, Nov 06 2012

[(Floor(n/3) + Ceiling(n/3)): n in [0..70]]; // Vincenzo Librandi, Aug 07 2011

A004396:=n->floor((2*n + 1)/3); seq(A004396(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[Floor[(2 n + 1)/3], {n, 0, 75}]
With[{n = 50}, Riffle[Range[0, n], Range[1, n, 2], {3, -1, 3}]] (* Harvey P. Dale, May 14 2015 *)
CoefficientList[Series[(x + x^3)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, Oct 27 2016 *)
a[ n_] := Quotient[2 n + 1, 3]; (* Michael Somos, Oct 23 2017 *)
a[ n_] := Sign[n] SeriesCoefficient[ (x + x^3) / ((1 - x) (1 - x^3)), {x, 0, Abs@n}]; (* Michael Somos, Oct 23 2017 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
f[-1]=0; f[n_]:=Length[Union[Plus@@@FrobeniusSolve[{1,2,3},n]]]; f/@Range[-1,100] (* Ivan N. Ianakiev, Mar 31 2025 *)

a(n)=2*n\/3 \\ Charles R Greathouse IV, Apr 17 2012

def a(n) : return( dimension_cusp_forms( Gamma0(3), 2*n+4) ); # Michael Somos, Jul 03 2014

G.f.: (x+x^3)/((1-x)*(1-x^3)).
a(n) = floor( (2*n + 1)/3 ).
a(n) = a(n-1) + (1/2)*((-1)^floor((4*n+2)/3) + 1), a(0) = 0. - Mario Catalani (mario.catalani(AT)unito.it), Oct 20 2003
a(n) = 2n/3 - cos(2*Pi*n/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9. - Paul Barry, Mar 18 2004
a(n) = A096777(n+1) - A096777(n) for n > 0. - Reinhard Zumkeller, Jul 09 2004
From Paul Barry, Jan 16 2005: (Start)
G.f.: x*(1+x^2)/(1-x-x^3+x^4).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
a(n) = Sum_{k = 0..n} binomial(n-k-1, k)*(-1)^k*A001045(n-2k). (End)
a(n) = (A006369(n) - (A006369(n) mod 2) * (-1)^(n mod 3)) / (1 + A006369(n) mod 2). - Reinhard Zumkeller, Jan 23 2005
a(n) = A004773(n) - A004523(n). - Reinhard Zumkeller, Aug 29 2005
a(n) = floor(n/3) + ceiling(n/3). - Jonathan Vos Post, Mar 19 2006
a(n+1) = A008620(2n). - Philippe Deléham, Dec 14 2006
a(A032766(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor((2*n^2+4*n+2)/(3*n+4)). - Gary Detlefs, Jul 13 2010
Euler transform of length 4 sequence [1, 1, 1, -1]. - Michael Somos, Jul 03 2014
a(n) = n - floor((n+1)/3). - Wesley Ivan Hurt, Sep 17 2015
a(n) = A092200(n) - floor((n+5)/3). - Filip Zaludek, Oct 27 2016
a(n) = -a(-n) for all n in Z. - Michael Somos, Oct 30 2016
E.g.f.: (2/9)*(3*exp(x)*x + sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 20 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Sep 29 2022

cp's OEIS Frontend

A168222 a(n) = A006369(A006369(n)).

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A168223 a(n) = A006369(n) - A006368(n).

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A349368 Dirichlet inverse of A006369, the inverse permutation of "amusical permutation".

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A349369 Sum of A006369 and its Dirichlet inverse, where A006369 is the inverse of "amusical permutation", A006368.

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A223084 Trajectory of 80 under the map n-> A006369(n).

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Maple

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A223085 Trajectory of 82 under the map n-> A006369(n).

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Maple

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A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

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A368228 Main diagonal of A368227: the n-th term in the trajectory of n under the A006369 map.

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A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

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