A131719 -id:A131719 - cp's OEIS frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 0

Paul Curtz, Jan 26 2008

Dirichlet series for the non-principal character modulo 6: L(s,chi) = Sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). - R. J. Mathar, Jul 31 2010

			G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
L. B. W. Jolley, Summation of Series, Dover (1961).

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.

&cat[[0, 1, 0, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
# Wesley Ivan Hurt, Jun 20 2016

a[ n_] := JacobiSymbol[-12, n]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 24 2014 *)
PadRight[{},120,{0,1,0,0,0,-1}] (* Harvey P. Dale, Aug 01 2021 *)

{a(n) = [0, 1, 0, 0, 0, -1][n%6+1]}; /* Michael Somos, Feb 10 2008 */

{a(n) = kronecker(-12, n)}; /* Michael Somos, Feb 10 2008 */

{a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))}; /* Michael Somos, Aug 11 2009 */

Euler transform of length 6 sequence [0, 0, 0, -1, 0, 1]. - Michael Somos, Feb 10 2008
G.f.: x * (1 - x^4) / (1 - x^6) = x*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos, Aug 11 2009
a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Aug 11 2009
a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.
sqrt(3)*a(n) = sin(Pi*n/3) + sin(2*Pi*n/3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - Ilya Gutkovskiy, Jun 21 2016

A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, 1763581, 4615823, 12082291, 31628466, 82798926, 216761547, 567474769, 1485645049, 3889431721, 10182603746, 26658304492, 69792188337, 182718064101, 478361686155, 1252366480135

Offset: 0

N. J. A. Sloane

Number of block fountains with exactly n coins in the base when mirror image fountains are identified. - Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010
a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - Ralf Stephan, Jul 06 2003
Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - Ralf Stephan, Jul 06 2003
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. - Herbert Kociemba, May 31 2004
The sequence 1,1,2,4,9,... has g.f. 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x))))=(1-3*x+x^2+x^2)/(1-4*x+3*x^2+2*x^3-x^4), and general term (A001519(n)+A000045(n+1))/2. It is the binomial transform of A001519 aerated. - Paul Barry, Dec 17 2009
The Kn3 and Kn4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence. - Johannes W. Meijer, Jul 14 2011
Convolution of [1,1,1,2,5,...], which is A001519 with another leading 1, and A212804. - R. J. Mathar, Apr 14 2018
a(n) is the number of Motzkin n-paths of height <= 3. - Alois P. Heinz, Nov 24 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..300 from Vincenzo Librandi)
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8, arXiv:math/0307050 [math.CO], 2003.
S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy)
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713 [math.CO], 2011.
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
Michael Woltermann, Problem 1826, Mathematics Magazine, 83 (2010), 304-305.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A001006, A001519, A131719.

A005207:=-(1-2*z-z^2+z^3)/(z^2-3*z+1)/(z^2+z-1); # Simon Plouffe in his 1992 dissertation with offset 0
a:= n-> (Matrix([[1,1,1,3]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4,-3,-2,1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..34); # Alois P. Heinz, Sep 06 2008

LinearRecurrence[{4, -3, -2, 1}, {1, 2, 4, 9}, 30] (* Jean-François Alcover, Jan 31 2016 *)

a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2

x='x+O('x^50); Vec(-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1))) \\ G. C. Greubel, Mar 05 2017

G.f.: 1-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1)).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
a(n) = (w^(2*n-1) + w^(1-2*n) + w^(n+1) - (-w)^(-1-n))/(4*w-2) where w = (1+sqrt(5))/2.
a(n) = (2/5)*Sum_{k=1..4} ( sin(Pi*k/5)^2*(1 + 2*cos(Pi*k/5))^n ). - Herbert Kociemba, May 31 2004
a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).
Let M = an infinite tridiagonal matrix with all 1's in the super and main diagonals and [1,1,1,0,0,0,...] in the subdiagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. - Gary W. Adamson, Jun 07 2011
2*a(n) = A000045(n+1) + A001519(n). - R. J. Mathar, Apr 14 2018
a(n) mod 2 = A131719(n+3). - Alois P. Heinz, Nov 24 2023

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

A103368 Period 6: repeat [1, 1, -1, -1, 0, 0].

Original entry on oeis.org

1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0

Offset: 0

Paul Barry, Feb 02 2005

The positive sequence is A131719(n+1) = a(n) = cos(2*Pi*n/3+Pi/3)/6 + sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 - sqrt(3)*cos(Pi*n/3+Pi/6)/6 + sin(Pi*n/3+Pi/6)/2 + 2/3, with g.f. (1+x^2) / ( (1-x)*(1-x+x^2)*(1+x+x^2) ).

Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A131719.

&cat [[1, 1, -1, -1, 0, 0]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A103368:=n->[1, 1, -1, -1, 0, 0][(n mod 6)+1]: seq(A103368(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

PadRight[{}, 100, {1, 1, -1, -1, 0, 0}] (* Wesley Ivan Hurt, Jun 20 2016 *)

G.f.: (1+x)/(1+x^2+x^4).
a(n) = Sum_{k=0..floor(n/2)} binomial(k, floor(n/2)-k)*(-1)^k.
a(n) = -cos(2*Pi*n/3+Pi/3)/2 + sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/2 + sin(Pi*n/3+Pi/6)/2.
a(n) = cos(Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: (sqrt(3)*sin(sqrt(3)*x/2) + 3*cos(sqrt(3)*x/2)*exp(x))*exp(-x/2)/3. - Ilya Gutkovskiy, Jun 21 2016

A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

Original entry on oeis.org

1, 1, 3, 8, 28, 107, 459, 2151, 10931, 59700, 348146, 2155925, 14112377, 97266301, 703484851, 5323515156, 42040470092, 345670438963, 2953171501547, 26166317121747, 240047041176843, 2276607815242880, 22290187889601330, 225018607554567149, 2339331996135377345

Offset: 1

Geoffrey Critzer, Apr 02 2013

nonn

			a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 08 2015
# second Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
      b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n-1, 1$2):
seq(a(n), n=1..25);  # Alois P. Heinz, May 17 2023

nn=25;Drop[Range[0,nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]],x],x],{x,0,nn}],x],1]

E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.
a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - Alois P. Heinz, Apr 02 2013
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022
From Alois P. Heinz, May 17 2023: (Start)
a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).
a(n) mod 2 = A131719(n+1). (End)

A131735 Period 6: repeat [0, 0, 1, 1, 1, 1].

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1

Offset: 0

Paul Curtz, Sep 19 2007

nonn

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

Cf. A131719.

PadRight[{},120,{0,0,1,1,1,1}] (* or *) LinearRecurrence[{1,-1,1,-1,1},{0,0,1,1,1},120] (* Harvey P. Dale, Jun 17 2015 *)

G.f.: -(x^2+1)*x^2/(x-1)/(x^2+x+1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007
a(n) = 2/3-(1/2)*cos((1/3)*Pi*n)-(1/6)*3^(1/2)*sin((1/3)*Pi*n)-(1/6)*cos((2/3)*Pi*n)-(1/6)*3^(1/2)*sin((2/3)*Pi*n). - R. J. Mathar, Nov 15 2007
a(n) = A131719(n-1), n>0. - R. J. Mathar, Jun 13 2008

A349788 Number of permutations of [n] having exactly one increasing cycle.

Original entry on oeis.org

0, 1, 1, 1, 5, 36, 234, 1597, 12459, 111451, 1116277, 12298958, 147655760, 1919465237, 26870436345, 403044639709, 6448695657957, 109628096021612, 1973308547820586, 37492874766408001, 749857477972731979, 15747006284752049759, 346434131946498886045

Offset: 0

Alois P. Heinz, Nov 30 2021

nonn

Cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < ... .

Exponential convolution of A000587 with A002627.

			a(4) = 5: (1)(243), (143)(2), (142)(3), (132)(4), (1234).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Column k=1 of A186754.

Cf. A000587, A002627, A131719, A186755, A186758.

b:= proc(n) option remember; series(`if`(n=0, 1, add((x+
     (j-1)!-1)*binomial(n-1, j-1)*b(n-j), j=1..n)), x, 2)
    end:
a:= n-> coeff(b(n), x, 1):
seq(a(n), n=0..23);

b[n_] := b[n] = Series[If[n == 0, 1, Sum[(x+
     (j-1)!-1)*Binomial[n-1, j-1]*b[n-j], {j, 1, n}]], {x, 0, 2}];
a[n_] := Coefficient[b[n], x, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

E.g.f.: exp(1-exp(x))*(exp(x)-1)/(1-x).
a(n) = A186758(n) - A186755(n).
a(n) = Sum_{j=0..n} binomial(n,j)*A000587(j)*A002627(n-j).
a(n) mod 2 = A131719(n).
a(n) ~ (exp(1) - 1) * exp(1 - exp(1)) * n!. - Vaclav Kotesovec, Dec 05 2021

A350175 Sum of the distinct block sizes over all partitions of [n].

Original entry on oeis.org

0, 1, 3, 13, 45, 196, 888, 4383, 22879, 129163, 768913, 4849912, 32202712, 224672241, 1640679589, 12517008985, 99484656169, 822410210044, 7055883373604, 62730142658947, 576984726864147, 5482889832932123, 53757450049841167, 543169144098559606, 5649499728403949184

Offset: 0

Alois P. Heinz, Jan 06 2022

nonn

			a(3) = 13 = 1*3 + 3*(1+2) + 1: 123, 1|23, 13|2, 12|3, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Wikipedia, Partition of a set

Cf. A000110, A070071, A131719, A132963.

b:= proc(n, i, c) option remember; `if`(n=0, c,
      `if`(i<1, 0, add(b(n-j*i, i-1, c+i*signum(j))*
      combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, c_] := b[n, i, c] = If[n == 0, c,
     If[i < 1, 0, Sum[b[n - j*i, i - 1, c + i*Sign[j]]*
     multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

a(n) mod 2 = A131719(n).

A363592 Number of partitions of [n] such that in each block the smallest element has the same parity as the largest element.

Original entry on oeis.org

1, 1, 1, 3, 6, 20, 55, 223, 761, 3595, 14532, 77818, 361605, 2155525, 11274781, 73822175, 428004750, 3046519516, 19348533739, 148493347507, 1023481273549, 8412534272415, 62450994058052, 546699337652602, 4343869829492281, 40308548641909593, 340994681344324137

Offset: 0

Alois P. Heinz, Jun 10 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 3: 123, 13|2, 1|2|3.
a(4) = 6: 123|4, 13|24, 13|2|4, 1|234, 1|24|3, 1|2|3|4.
a(5) = 20: 12345, 1235|4, 123|4|5, 1245|3, 125|3|4, 1345|2, 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|234, 1|234|5, 145|2|3, 15|24|3, 1|24|35, 1|24|3|5, 1|2|345, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..150
Wikipedia, Partition of a set

Cf. A000110, A131719, A361084.

b:= proc(n, x, y, u, v) option remember; `if`(y+u>n, 0, `if`(n=0, 1,
      `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+
      `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x)))
    end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..30);

a(n) mod 2 = A131719(n+1).

A283439 Hankel transform of A033434.

Original entry on oeis.org

1, -3, -9, -6, 10, 25, 15, -21, -49, -28, 36, 81, 45, -55, -121, -66, 78, 169, 91, -105, -225, -120, 136, 289, 153, -171, -361, -190, 210, 441, 231, -253, -529, -276, 300, 625, 325, -351, -729, -378, 406, 841, 435, -465, -961, -496, 528, 1089, 561, -595, -1225

Offset: 0

Paul Barry, Mar 07 2017

a(n) modulo 2 is A131719(n+2).

Cf. A131719, A033434.

CoefficientList[Series[(1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3) ,{x, 0, 35}], x] (* Indranil Ghosh, Mar 08 2017 *)

print(Vec((1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3) + O(x^36))); \\ Indranil Ghosh, Mar 08 2017

G.f.: (1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3).
a(3*k)     =  (-1)^k*(k + 1)*(2*k + 1).
a(3*k + 1) = -(-1)^k*(k + 1)*(2*k + 3).
a(3*k + 2) = -(-1)^k*(k + 3)^2.

More terms from Indranil Ghosh, Mar 08 2017

cp's OEIS Frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

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A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

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A103368 Period 6: repeat [1, 1, -1, -1, 0, 0].

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A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

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A131735 Period 6: repeat [0, 0, 1, 1, 1, 1].

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A349788 Number of permutations of [n] having exactly one increasing cycle.

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Formula

A350175 Sum of the distinct block sizes over all partitions of [n].

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