A142150 -id:A142150 - cp's OEIS frontend

A175954 Unlabeled (cyclic) Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations of the circle.

1, 1, 2, 2, 4, 5, 12, 19, 46, 95, 230, 528, 1320, 3219, 8172, 20714, 53478, 138635, 363486, 957858, 2543476, 6788019, 18218772, 49120019, 133036406, 361736109, 987316658, 2703991820, 7429445752, 20473889133, 56579632732, 156766505691

Offset: 0

Max Alekseyev, Oct 29 2010

nonn

Unlabeled version of A001006.

The number of such chord configurations on 2n vertices with n chords is given by A002995(n+1).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Andrew Howroyd, Chord Configuration Symmetries

Cf. A001006, A185100, A002426, A002995.

a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n-1] + Sum[a1006[k]* a1006[n -2-k], {k, 0, n-2}];
a142150[n_] := n*(1 + (-1)^n)/4;
a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
a[0] = 1; a[n_] := (1/n)*(a1006[n]+a142150[n]*a1006[n/2-1] + Sum[EulerPhi[ n/d]*a2426[d], {d, Most @ Divisors[n]}]);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

For odd prime p, a(p) = (A001006(p) - 1)/p + 1.

a(n) = (1/n) * (A001006(n) + A142150(n) * A001006(n/2-1) + Sum{d|n, dA002426(d)). - Andrew Howroyd, Apr 01 2017

A185100 Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 11, 16, 36, 65, 150, 312, 756, 1743, 4353, 10732, 27489, 70379, 183866, 481952, 1277784, 3402661, 9126689, 24584870, 66567924, 180939737, 493801694, 1352203202, 3715137460, 10237545525, 28291018283, 78384998904, 217715672036, 606103034821, 1691020991782, 4727601528674, 13242641322252, 37162431389051, 104469244613429

Offset: 0

Max Alekseyev, Feb 07 2011

nonn

Unlabeled version of A001006. Another version is given by A175954.

The number of ways of drawing exactly n chords joining 2n unlabeled points up to rotations and reflections is A006082(n+1). - Andrey Zabolotskiy, May 24 2018

Andrew Howroyd, Table of n, a(n) for n = 0..200
Andrew Howroyd, Chord Configuration Symmetries

Cf. A001006 (labeled points), A175954 (up to rotations only), A175955, A005773, A006082.

a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];
a142150[n_] := n*(1 + (-1)^n)/4;
a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
a175954[0] = 1; a175954[n_] := (1/n)*(a1006[n] + a142150[n]*a1006[n/2 - 1] + Sum[EulerPhi[n/d]*a2426[d], {d, Most @Divisors[n]}]);
a5773[0] = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
a[0] = 1;
a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];
a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

a(2n+1) = (1/2) * (A175954(2n+1) + A005773(n+1)). - Andrew Howroyd, Apr 01 2017

a(2n) = (1/4) * (2 * A175954(2n) + A005773(n) + A005773(n+1) + A001006(n-1)) for n > 0. - Andrew Howroyd, Apr 01 2017

A225972 The number of binary pattern classes in the (2,n)-rectangular grid with 3 '1's and (2n-3) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

0, 0, 1, 6, 14, 32, 55, 94, 140, 208, 285, 390, 506, 656, 819, 1022, 1240, 1504, 1785, 2118, 2470, 2880, 3311, 3806, 4324, 4912, 5525, 6214, 6930, 7728, 8555, 9470, 10416, 11456, 12529, 13702, 14910, 16224, 17575, 19038, 20540, 22160, 23821, 25606, 27434, 29392

Offset: 0

Yosu Yurramendi, May 26 2013

Also the edge count of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Edge Count
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A226048, A000330.

Cf. A289179 (edge count of white bishop graph).

[(1/4)*(Binomial(2*(n-1),3)+2*Binomial(n-2,1)*(1/2)*(1+(-1)^n)): n in [1..50]]; // Vincenzo Librandi, Sep 04 2013

A225972:=n->(n-1)*(4*n^2-2*n-3*(-1)^n+3)/12; seq(A225972(n), n=0..40); # Wesley Ivan Hurt, Mar 02 2014

Table[(n - 1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12, {n, 0, 40}] (* Bruno Berselli, May 29 2013 *)
CoefficientList[Series[x^2 (1 + 4 x + x^2 + 2 x^3) / ((1 + x)^2 (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 04 2013 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 6, 14, 32, 55}, 20] (* Eric W. Weisstein, Jun 27 2017 *)

a <- vector()
    for(n in 0:40) a[n] <- (1/4)*(choose(2*(n-1),3) + 2*choose(n-2,1)*(1/2)*(1+(-1)^n))
    a  # Yosu Yurramendi and María Merino, Aug 21 2013

a(n) = A000330(n) + A142150(n) = (n-1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) with n > 5, a(0)=0, a(1)=0, a(2)=1, a(3)=6, a(4)=14, a(5)=32.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 4*(n-4)*(-1)^n with n > 3, a(0)=0, a(1)=0, a(2)=1, a(3)=6.
G.f.: x^2*(1 + 4*x + x^2 + 2*x^3)/((1+x)^2*(1-x)^4). - Bruno Berselli, May 29 2013
a(n) = (1/4)*(binomial(2*(n-1),3) + 2*binomial(n-2,1)*(1/2)*(1+(-1)^n)). - Yosu Yurramendi and María Merino, Aug 21 2013
a(n) = A005993(n-2) + A199771(n-1), n >= 2. - Christopher Hunt Gribble, Mar 02 2014

More terms from Vincenzo Librandi, Sep 04 2013

A301283 Coordination sequence for node of type V1 in "car" 2-D tiling (or net).

Original entry on oeis.org

1, 3, 6, 8, 12, 17, 18, 20, 26, 29, 29, 33, 39, 41, 41, 45, 52, 54, 52, 57, 66, 66, 63, 70, 79, 78, 75, 82, 92, 91, 86, 94, 106, 103, 97, 107, 119, 115, 109, 119, 132, 128, 120, 131, 146, 140, 131, 144, 159, 152, 143, 156, 172, 165, 154, 168, 186, 177, 165

Offset: 0

N. J. A. Sloane, Mar 23 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The car tiling (or net)
Rémy Sigrist, PARI program for A301283
Rémy Sigrist, Illustration of first terms

Cf. A301285.

PARI
```
See Links section.
```

Conjectures from Colin Barker, Mar 30 2018: (Start)
G.f.: (1 + 2*x + 5*x^2 + 5*x^3 + 9*x^4 + 6*x^5 + 6*x^6 + 3*x^7 + x^8 - x^10) / ((1 - x)^2*(1 + x^2)^2*(1 + x + x^2)).
a(n) = a(n-1) - 2*a(n-2) + 3*a(n-3) - 2*a(n-4) + 3*a(n-5) - 2*a(n-6) + a(n-7) - a(n-8) for n>8.
(End)
Equivalent conjecture: 12*a(n) = 37*n+4*b(n)+6*(-1)^(n/2)*A142150(n+2)+3*c(n) for n>2, where b(n)=0,-1,1 (3-periodic, n>=0) and c(n) = -6,5,6,-5 (4-periodic, n>=0). - R. J. Mathar, Mar 31 2018

More terms from Rémy Sigrist, Mar 28 2018

A175922 Period 5: repeat [1, 1, 2, -1, 2].

Original entry on oeis.org

1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2, 1, 1, 2, -1, 2

Offset: 1

Jaroslav Krizek, Oct 17 2010

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A040001, A028356, A142150, A000012, A175921.

Flat(List([1..30],n->[1,1,2,-1,2])); # Muniru A Asiru, Sep 28 2018

&cat[[1, 1, 2, -1, 2]^^20]; // Vincenzo Librandi, Sep 28 2018

seq(op([1,1,2,-1,2]),n=1..30); # Muniru A Asiru, Sep 28 2018

PadRight[{}, 100, {1, 1, 2, -1, 2}] (* Vincenzo Librandi, Sep 28 2018 *)

a(n)=[2,1,1,2,-1][n%5+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 1 + (2/5)*(cos(2*n*Pi/5) + cos(4*n*Pi/5) - 2*cos(2*(n+1)*Pi/5) - sin((4*n+3)*Pi/10) + 2*sin((8*n+3)*Pi/10) + sin((8*n+1)*Pi/10)). - Wesley Ivan Hurt, Sep 27 2018
G.f.: x*(1 + x + 2*x^2 - x^3 + 2*x^4) / (1 - x^5). - Vincenzo Librandi, Sep 28 2018
a(n) = a(n-5). - Wesley Ivan Hurt, Jun 25 2022

Edited by Joerg Arndt, Sep 16 2013

A257845 a(n) = floor(n/5) * (n mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 0, 6, 12, 18, 24, 0, 7, 14, 21, 28, 0, 8, 16, 24, 32, 0, 9, 18, 27, 36, 0, 10, 20, 30, 40, 0, 11, 22, 33, 44, 0, 12, 24, 36, 48, 0, 13, 26, 39, 52, 0, 14, 28, 42, 56

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 5, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,0,0,0,0,0,1,2,3,4},80] (* Harvey P. Dale, Aug 15 2021 *)

PARI
```
a(n,b=5)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0,0], Vec(x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-5)-a(n-10). - Colin Barker, May 11 2015

G.f.: x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A257849 a(n) = floor(n/9) * (n mod 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 2, 4, 6, 8, 10, 12, 14, 16, 0, 3, 6, 9, 12, 15, 18, 21, 24, 0, 4, 8, 12, 16, 20, 24, 28, 32, 0, 5, 10, 15, 20, 25, 30, 35, 40, 0, 6, 12, 18, 24, 30, 36, 42, 48, 0, 7, 14, 21, 28, 35, 42, 49, 56, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 9, multiply the last digit by the number with its last digit removed.

See A142150(n-1) for the base 2 analog, and A115273, A257844 - A257850 for the base 3 - base 10 variants.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

[Floor(n/9)*(n mod 9): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/9] Mod[n, 9], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
A257849(n)=n\9*(n%9)
```

concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

from math import prod
def A257849(n): return prod(divmod(n,9)) # Chai Wah Wu, Jan 19 2023

[floor(n/9)*(n % 9)  for n in (0..80)]; # Bruno Berselli, May 11 2015

a(n) = 2*a(n-9)-a(n-18). - Colin Barker, May 11 2015

G.f.: x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2). - Colin Barker, May 11 2015

A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2).

Original entry on oeis.org

1, 0, 0, -1, 2, 20, -120, -4608, 41952, 2325024, -34876800, -3133087200, 66120252480, 8258565859200, -239533775631360, -40631838221721600, 1532513262269767680, 335620705700380262400, -16054693916748370329600, -4428138916386119015424000, 261291002534430572648448000

Offset: 0

Stefano Spezia, Jan 04 2022

sign

The matrix M(n) is the n-th principal submatrix of the array A010751.
In the n X n matrix M(n): the zero element appears 2*n - 1 times; the positive integers k appears iff 0 < k < floor(n/2), 2*n - 1 - A040002(k-1) times; the negative integer k appears iff -k < ceiling(n/2),  2*n - 5 + 4*(k + 1) times.
det(M(n)) = 0, except for n = 3 for which det(M(3)) = -1.
The trace and the subdiagonal sum of the matrix M(n) are zero.
The antitrace of the matrix M(n) is A142150(n+1).
The superdiagonal sum of the matrix M(n) is equal to n - 1.
The sum of the elements of the matrix M(n) is A002620(n).

			For n = 3 the matrix M(3) is
     0, 1, 1
     0, 0, 1
    -1, 0, 0
with permanent a(3) = -1.
For n = 4 the matrix M(4) is
    0,  1,  1,  2
    0,  0,  1,  1
   -1,  0,  0,  1
   -1, -1,  0,  0
with permanent a(4) = 2.

Cf. A002620, A004526, A010751, A040002, A060747, A110654, A142150.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> floor((j-i+1)/2)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2022

Join[{1},Table[Permanent[Table[Floor[(j-i+1)/2],{i,n},{j,n}]],{n,20}]]

a(n) = matpermanent(matrix(n, n, i, j, (j - i + 1)\2)); \\ Michel Marcus, Jan 04 2022

from sympy import Matrix
def A350549(n): return 1 if n == 0 else Matrix(n,n,lambda i,j:(j-i+1)//2).per() # Chai Wah Wu, Jan 12 2022

A131360 a(4n) = a(4n+1) = 0, a(4n+2) = 2n, a(4n+3) = 2n+1.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Sep 30 2007

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

Cf. A142150, A021913.

Array[Floor[(# - 1)/2] Floor[Mod[#, 4]/2] &, 88, 0] (* Michael De Vlieger, Sep 22 2021 *)

concat(vector(3), Vec(x^3*(x^3+x^2-x+1)/((x-1)^2*(x+1)*(x^2+1)^2) + O(x^100))) \\ Colin Barker, Jul 01 2015

G.f.: x^3*(x^3+x^2-x+1) / ((x-1)^2*(x+1)*(x^2+1)^2). - Colin Barker, Jul 01 2015
a(n) = (cos(n*Pi/2)+sin(n*Pi/2)-1)*((2n-3)*cos(n*Pi/2)+cos(n*Pi)+(2n-3)*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Sep 24 2017
a(n) = floor((n-1)/2)*A021913(n). - Lechoslaw Ratajczak, Sep 22 2021

A175921 Period 5: repeat [1, 2, 2, -1, 1].

Original entry on oeis.org

1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1, 1, 2, 2, -1, 1

Offset: 1

Jaroslav Krizek, Oct 17 2010

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

Cf. A040001, A028356, A142150, A000012, A175922.

PadRight[{},120,{1,2,2,-1,1}] (* Harvey P. Dale, Dec 30 2016 *)

a(n)=[1,1,2,2,-1][n%5+1] \\ Charles R Greathouse IV, Mar 12 2017

Edited by Joerg Arndt, Sep 16 2013

cp's OEIS Frontend

A175954 Unlabeled (cyclic) Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations of the circle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A185100 Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A225972 The number of binary pattern classes in the (2,n)-rectangular grid with 3 '1's and (2n-3) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

R

Formula

Extensions

A301283 Coordination sequence for node of type V1 in "car" 2-D tiling (or net).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A175922 Period 5: repeat [1, 1, 2, -1, 2].

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

A257845 a(n) = floor(n/5) * (n mod 5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A257849 a(n) = floor(n/9) * (n mod 9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma