A047749 -id:A047749 - cp's OEIS frontend

A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3.

1, 1, 1, 1, 2, 3, 4, 9, 15, 22, 52, 91, 140, 340, 612, 969, 2394, 4389, 7084, 17710, 32890, 53820, 135720, 254475, 420732, 1068012, 2017356, 3362260, 8579560, 16301164, 27343888, 70068713, 133767543, 225568798, 580034052, 1111731933, 1882933364, 4855986044, 9338434700

Offset: 0

Paul Barry, Nov 06 2006

Row sums of Riordan array (1,x(1-x^3))^(-1). Also row sums of A124752.
a(n) is the number of ordered trees (A000108) with n vertices in which every non-leaf non-root vertex has exactly two children that are leaves. For example, a(4) counts the 2 trees
  \ /
   |   and  \|/ . - David Callan, Aug 22 2014

J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

Cf. A084080, A002293, A069271 (trisection), A006632 (trisection).

Cf. A047749, A118968.

A124753 := proc(n)
    local k,np;
    k := modp(n,3) ;
    np := floor(n/3) ;
    (k+1)*binomial(np+n,np)/(n+1) ;
end proc:
seq(A124753(n),n=0..40) ; # R. J. Mathar, Oct 30 2014

a[n_] := Module[{q, k}, {q, k} = QuotientRemainder[n, 3]; (k+1)*Binomial[4q + k, q]/(3q + k + 1)];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 20 2017 *)

{a(n)=local(A=1+x); for(i=1,n,A=1+x*A*exp(sum(m=1,n\3,3*polcoeff(log(A+x*O(x^n)),3*m)*x^(3*m))+x*O(x^n))); polcoeff(A,n)} \\ Paul D. Hanna, Jun 04 2012

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\3, 4, n%3+1); \\ Seiichi Manyama, Jul 20 2025

a(3n) = A002293(n), a(3n+1) = A069271(n), a(3n+2) = A006632(n+1).
a(n) = ((mod(n,3)+1)*C(4*floor(n/3)+mod(n,3), floor(n/3))/ (3*floor(n/3) + 1 + mod(n, 3))). - Paul Barry, Dec 14 2006
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(w*x)*A(w^2*x), where w = exp(2*Pi*I/3). - Paul D. Hanna, Jun 04 2012
G.f. satisfies: A(x) = 1 + x*A(x)*G(x^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. - Paul D. Hanna, Jun 04 2012
Conjecture: +8019*n*(n-1)*(n+1)*a(n) +17496*n*(n-1)*(n-3)*a(n-1) +2592*(3*n-5)*(n-1)*(3*n-16)*a(n-2) +216*(-224*n^3+48*n^2+3926*n-6331)*a(n-3) +576*(-288*n^3+2448*n^2-6558*n+5443)*a(n-4) +768*(-288*n^3+3600*n^2-14878*n+20375)*a(n-5) -8192*(4*n-23)*(2*n-11)*(4*n-21)*a(n-6)=0. - R. J. Mathar, Oct 30 2014
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} a(3*k) * a(n-1-3*k). - Seiichi Manyama, Jul 07 2025

A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 6, 1, 1, 1, 1, 3, 5, 7, 10, 1, 1, 1, 1, 4, 5, 16, 12, 20, 1, 1, 1, 1, 4, 7, 18, 31, 30, 35, 1, 1, 1, 1, 5, 7, 31, 35, 102, 55, 70, 1, 1, 1, 1, 5, 9, 34, 64, 136, 213, 143, 126, 1

Offset: 0

Andrew Howroyd, Feb 07 2024

T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.

			Array begins:
===============================================
n\k| 1  2   3   4    5    6    7    8     9 ...
---+-------------------------------------------
0  | 1  1   1   1    1    1    1    1     1 ...
1  | 1  1   1   1    1    1    1    1     1 ...
2  | 1  1   1   1    1    1    1    1     1 ...
3  | 1  2   2   3    3    4    4    5     5 ...
4  | 1  3   3   5    5    7    7    9     9 ...
5  | 1  6   7  16   18   31   34   51    55 ...
6  | 1 10  12  31   35   64   70  109   117 ...
7  | 1 20  30 102  136  296  368  651   775 ...
8  | 1 35  55 213  285  663  819 1513  1785 ...
9  | 1 70 143 712 1155 3142 4495 9304 12350 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
Wikipedia, Fuss-Catalan number.
Wikipedia, Noncrossing partition.

Columns are: A000012, A001405(n-1), A047749 (k=3), A369930 (k=4), A143546 (k=5), A143547 (k=7), A143554 (k=9), A192893 (k=11).

Cf. A070914, A303694, A303929, A361236, A361239, A370060, A370062.

\\ u(n,k,r) are Fuss-Catalan numbers.
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
T(n, k)={if(n==0, 1, if(k%2, if(n%2, 2*u(n\2, k, (k+1)/2), u(n/2, k, 1) + u(n/2-1, k, k)), e(n, k) + if(n%2, u(n\2, k, k/2)))/2)}

T(n,k) = 2*A303929(n,k) - A303694(n,k).

T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).

A047765 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type P.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 0, 7, 0, 12, 0, 29, 0, 55, 0, 143, 0, 271, 0, 728, 0, 1428, 0, 3873, 0, 7752, 0, 21318, 0, 43256, 0, 120175, 0, 246675, 0, 690678, 0, 1430715, 0, 4032015, 0, 8414610, 0, 23841480, 0, 50067108, 0, 142498637, 0, 300830572

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type P achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral face (21); the order of the symmetry group is 4. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047765 examples

Cf. A047767.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047764 (type Q).

Table[If[OddQ[n],0,If[OddQ[n/2],2Binomial[(3n-2)/4,(n-2)/4],Binomial[3n/4,n/4]]/(n/2+1)-Switch[Mod[n,12],2,6Binomial[(n-2)/4,(n-2)/12],8,12Binomial[(n-4)/4,(n-2)/6],,0]/(n+4)],{n,52}] (* _Robert A. Russell, Mar 22 2024 *)

If n=2m then A047749(m) - A047764(n), otherwise 0.

G.f.: G(z^4) + z^2*G(z^4)^2 - z^2*G(z^12) - z^8*G(z^12)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 3, 0, 3, 5, 0, 7, 11, 0, 12, 23, 0, 30, 55, 0, 55, 114, 0, 143, 272, 0, 273, 588, 0, 728, 1428, 0, 1428, 3156, 0, 3876, 7750, 0, 7752, 17427, 0, 21318, 43263, 0, 43263, 98516, 0, 120175, 246672, 0, 246675, 567281, 0

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type D achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral cell (32); the order of the symmetry group is 6. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 23 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047773 examples

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047750 (type V), A047751 (type K), A047764 (type Q).

Table[Switch[Mod[n,6],1,If[1==n,0,3Binomial[(n-1)/2,(n-1)/6]/(n+2)],2,6Binomial[n/2,(n-2)/6]/(n+4)-3Binomial[(n-2)/2,(n-2)/6]/(2n+2)-If[2==Mod[n,12],3Binomial[(n-2)/4,(n-2)/12],6Binomial[(n-4)/4,(n-8)/12]]/(n+4),4,6Binomial[(n-2)/2,(n-4)/6]/(n+2),5,3Binomial[(n+1)/2,(n+1)/6]/(n+4)-Switch[Mod[n,24],5,12Binomial[(n-5)/8,(n-5)/24],17,24Binomial[(n-9)/8,(n-17)/24],,0]/(n+7),,0],{n,60}] (* Robert A. Russell, Mar 23 2024 *)

/* here U=A047749, V=A047750, K=A047751, and Q=A047764 */
U(n)={if(n%2,my(m=(n-1)/2);(3*m+1)!/((m+1)!*(2*m+1)!),my(m=n/2);(3*m)!/(m!*(2*m+1)!))};
V(n)={if(n%2,my(m=(n-1)/2);6*(3*m+2)!/(m!*(2*m+3)!),my(m=n/2);(3*m)!*(5*m+1)/((m+1)!*(2*m+1)!))};
K(n)={if(n==1,1,if(n<5,0,if(n%12==5,U((n-5)/12),0)))};
Q(n)={if(n<8,0,if(n%6==2,U((n-2)/6),0))};
D(n)={if(n<3||n%3==0,0,if(n%3==1,U((n-1)/3),(1/2)*(V((n-2)/3)-2*K(n)-Q(n))))};
for(k=1,57,print1(D(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

If n=3m+2 then (1/2)*(A047750(m) - 2*A047751(n) - A047764(n)), if n=3m+1 then A047749(m), otherwise 0.

G.f.: (G(z^6)-1)/z + z*G(z^6) - z + z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^5*G(z^24) - z^17*G(z^24)^2 - (z^2*G(z^6) + z^2*G(z^12) + z^8*G(z^12)^2)/2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 23 2024

A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 52, 140, 340, 969, 2394, 7084, 17710, 53820, 135720, 420732, 1068012, 3362260, 8579560, 27343888, 70068713, 225568798, 580034052, 1882933364, 4855986044, 15875338990, 41043559340, 134993766600

Offset: 1

Robert A. Russell, Jan 23 2024

A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Column k=5 of A370060.

Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A002293 (rooted), A047749 {4,oo}, A143546 {6,oo}.

p=5; Table[If[EvenQ[n],Binomial[(p-1)n/2,n/2]/((p-2)n/2+1),If[OddQ[p],(p-1)Binomial[(p-1)n/2-1,(n-1)/2]/((p-2)n+1),p Binomial[(p-1)n/2-1/2,(n-1)/2]/((p-2)n+2)]],{n,35}]

For n even, a(n) = C(2n,n/2)/(3n/2+1).
For n odd, a(n) = 4*C(2n-1,(n-1)/2)/(3n+1).
a(n+2)/a(n) ~ 256/27. a(2m+1)/a(2m) ~ 32/9; a(2m)/a(2m-1) ~ 8/3.
a(n) = 2*A005040(n) - A005038(n) = A005038(n) - 2*A369471(n) = A005040(n) - A369471(n).
G.f.: G(z^2)+z*G(z^2)^2, where G(z)=1+z*G(z)^4, the generating function for A002293.
a(2m) = A002293(m) ~ (4^4/3^3)^m*sqrt(4/(2*Pi*(3*m)^3)). - Robert A. Russell, Jul 15 2024

A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 7, 5, 1, 1, 3, 5, 9, 12, 5, 1, 1, 4, 6, 18, 22, 30, 14, 1, 1, 4, 7, 21, 35, 52, 55, 14, 1, 1, 5, 8, 34, 51, 136, 140, 143, 42, 1, 1, 5, 9, 38, 70, 190, 285, 340, 273, 42, 1, 1, 6, 10, 55, 92, 368, 506, 1155, 969, 728, 132

Offset: 1

Andrew Howroyd, Feb 08 2024

The polygon prior to dissection will have n*(k-2)+2 sides.

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   2   2    3    3    4    4     5 ...
4  |  2   3   4    5    6    7    8     9 ...
5  |  2   7   9   18   21   34   38    55 ...
6  |  5  12  22   35   51   70   92   117 ...
7  |  5  30  52  136  190  368  468   775 ...
8  | 14  55 140  285  506  819 1240  1785 ...
9  | 14 143 340 1155 1950 4495 6545 12350 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns are A208355(n-1), A047749 (k=4), A369472 (k=5), A143546 (k=6), A143547 (k=8), A143554 (k=10), A192893 (k=12).

Cf. A070914 (rooted), A295224 (oriented), A295260 (unoriented), A369929, A370060 (achiral rooted at cell).

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {(if(n%2, u((n-1)/2, k, k\2), if(k%2, u(n/2-1, k, k-1), u(n/2, k, 1))))}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

T(n,k) = 2*A295260(n,k) - A295224(n,k).
T(n,2*k+1) = A370060(n,2*k+1).
T(n,2*k) = A369929(n,2*k-1).

A047762 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type E.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 6, 0, 32, 0, 176, 0, 952, 0, 5302, 0, 29960, 0, 172536, 0, 1007575, 0, 5959656, 0, 35622384, 0, 214875104, 0, 1306303424, 0, 7995896502, 0, 49236826080, 0, 304799714960, 0, 1895785216039, 0, 11841367945110, 0, 74245791718824

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type E chiral symmetry and n tetrahedral cells. The axis of symmetry connects opposite edge centers of a tetrahedron (31); the order of the symmetry group is 2. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047762 example.

Cf. A047749, A047763.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K), A047752 (type J), A047753 (type I), A047754 (type H), A047758 (type G), A047760 (type F).

Table[If[OddQ[n],Binomial[(3n-1)/2,(n-1)/2]/(n+1)-If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/4]/((n+1))+(4Binomial[(3n+1)/4,(n-1)/4]-If[1==Mod[n,8],4Binomial[(3n-3)/8,(n-1)/8],8Binomial[(3n-7)/8,(n-5)/8]])/(n+3),2Binomial[(3n+3)/4,(n+1)/4]/(n+3)],0]/2,{n,40}] (* Robert A. Russell, Mar 22 2024 *)

If n=2m+1 then (1/4)*(A047749(n) - 2*A047760(n) - 6*A047758(n) - 2*A047754(n) - 3*A047753(n) - 2*A047752(n) - A047751(n)), otherwise 0.

G.f.: (z^2*G(z^2)^2 - (2+z^2)*G(z^4) - 2*z^2*G(z^4)^2 + 2*(1 + z^2*G(z^8) + z^6*G(z^8)^2)) / (4*z), where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A047774 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type C.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 5, 6, 0, 26, 32, 0, 133, 176, 0, 708, 952, 0, 3861, 5302, 0, 21604, 29960, 0, 123266, 172535, 0, 715221, 1007575, 0, 4206956, 5959656, 0, 25032840, 35622384, 0, 150413348, 214875099, 0, 911379384, 1306303424, 0, 5562367173

Offset: 1

N. J. A. Sloane

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type C chiral symmetry and n tetrahedral cells. The axis of rotational symmetry is the altitude of a tetrahedral cell (32); the order of the symmetry group is 3. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 25 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047774 examples.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047749 (type U), A047751 (type K), A047752 (type J), A047764 (type Q), A047766 (types N|O), A047773 (type D).

# T=A001764
T := proc(n)
    if n < 0 then
        0;
    else
        (3*n)!/n!/(2*n+1)! ;
    end if;
end proc:
# U=A047749
U := proc(n)
    if type(n,'integer') then
        if type(n,'even') then
            T(n/2) ;
        else
            (3*n-1)/(n+1)*T((n-1)/2) ;
        end if;
    else
        0 ;
    end if;
end proc:
# V=A047750
V := proc(n)
    if type(n,'integer') then
        if type(n,'even') then
            2*U(n+1)-U(n) ;
        else
            2*U(n+1) ;
        end if;
    else
        0;
    end if;
end proc:
K := proc(n)
    if n < 1 then
        0 ;
    elif n = 1 then
        1;
    else
        U((n-5)/12) ;
    end if;
end proc:
J := proc(n)
    if type((n-5)/12,'integer') then
        T((n-5)/12)-K(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
Q := proc(n)
    if type((n-2)/6,'integer') then
        U((n-2)/6) ;
    else
        0 ;
    end if;
end proc:
N := proc(n)
    if type((n-2)/6,'integer') then
        T((n-2)/6)-Q(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
DD := proc(n)
    2*U((n-1)/3)+V((n-2)/3)-2*K(n)-Q(n) ;
    %/2 ;
end proc:
OO := proc(n)
    if type((n-2)/6,'integer') then
        T((n-2)/6)-Q(n) ;
        %/2 ;
    else
        0;
    end if ;
end proc:
C := proc(n)
    if n = 1 then
        0;
      elif modp(n,3) = 1 then
        T((n-1)/3)-DD(n) ;
        %/2 ;
    else
        U((2*n-1)/3)-2*DD(n)-4*J(n) -2*K(n)-2*N(n)-2*OO(n)-Q(n) ;
        %/4 ;
    end if;
end proc:
seq(C(n),n=1..50) ; # R. J. Mathar, Jul 10 2013

t[n_?IntegerQ] := Binomial[3 n, n] / (2 n + 1); t[_] = 0;
u[n_] := t[n/2] + ((3n-1)/(n+1)) t[(n-1)/2];
c[n_] := (2 (t[(n-1)/3] - u[(n-1)/3] - u[(n+1)/3] + u[(n-2)/6] + u[(n-5)/12] - t[(n-5)/12]) + u[(2n-1)/3] - t[(n-2)/6]) / 4;
Array[c, 46] (* Andrey Zabolotskiy, Jul 30 2023 and Apr 03 2024, using R. J. Mathar's code above *)
Table[(If[2==Mod[n,3],3Binomial[n-1,(n-2)/3]-If[2==Mod[n,6],3Binomial[(n-2)/2,(n-2)/6],0],0]/(2n+2)-Switch[Mod[n,3],1,If[1==Mod[n,6],3Binomial[(n-1)/2,(n-1)/6],6Binomial[(n-2)/2,(n-4)/6]]/(n+2)-3Binomial[n-1,(n-1)/3]/(2n+1),2,If[2==Mod[n,6],6Binomial[n/2,(n-2)/6]-If[2==Mod[n,12],6Binomial[(n-2)/4,(n-2)/12],12Binomial[n/4-1,(n-8)/12]],3Binomial[(n+1)/2,(n+1)/6]]/(n+4),,0]-If[5==Mod[n,12],6Binomial[(n-5)/4,(n-5)/12]/(n+1)-If[5==Mod[n,24],12Binomial[(n-5)/8,(n-5)/24],24Binomial[(n-9)/8,(n-17)/24]]/(n+7),0])/2,{n,50}] (* _Robert A. Russell, Mar 25 2024 *)

From Robert A. Russell, Mar 25 2024: (Start)
a(n) = (2*A001764((n-1)/3) + A047749((2n-1)/3) - 2*A047773(n) - 4*A047752(n) - 2*A047751(n) - 4*A047766(n) - A047773(n)) / 4.
G.f.: ((1 - G(z^6))/z + z^2*(G(z^3)^2 - G(z^6))/2 + z*G(z^3) - z*G(z^6) - z^2*G(z^6)^2 - z^4*G(z^6)^2 + z^2*G(z^12) - z^5*G(z^12) + z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

More terms from R. J. Mathar, Jul 10 2013

A047775 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type B.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 66, 131, 349, 708, 1911, 3856, 10604, 21597, 59961, 123266, 345060, 715198, 2015416, 4206926, 11919257, 25032840, 71246129, 150413234, 429750208, 911379241, 2612614298, 5562367173, 15991792731, 34164355260

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type B achiral symmetry and n tetrahedral cells. The plane of symmetry bisects a tetrahedral cell (321); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 29 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.
Robert A. Russell, Mathematica Graphics3D program for A047775 examples

Cf. A047772.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047751 (type K), A047753 (type I), A047760 (type F), A047764 (type Q), A047765 (type P), A047773 (type D).

Table[If[n<5,0,If[OddQ[n],2Binomial[(3n-1)/2,(n-1)/2]/(n+1)+If[1==Mod[n,4],2Binomial[3(n-1)/4,(n-1)/4]/(n+1),0],Binomial[3n/2,n/2]/(n+1)-If[OddQ[n/2],4Binomial[(3n-2)/4,(n-2)/4],2Binomial[3n/4,n/4]]/(n+2)]/2+If[2==Mod[n,6],3Binomial[(n-2)/2,(n-2)/6]/(n+1)+If[2==Mod[n,12],6Binomial[(n-2)/4,(n-2)/12],12Binomial[n/4-1,(n-8)/12]]/(n+4),0]/2-Switch[Mod[n,4],1,4Binomial[(3n+1)/4,(n-1)/4],3,2Binomial[(3n+3)/4,(n+1)/4],,0]/(n+3)-Switch[Mod[n,6],1,3Binomial[(n-1)/2,(n-1)/6]/(n+2),2,6Binomial[n/2,(n-2)/6]/(n+4),4,6Binomial[(n-2)/2,(n-4)/6]/(n+2),,0]-If[5==Mod[n,6],3Binomial[(n+1)/2,(n+1)/6]/(n+4)-Switch[Mod[n,24],5,12Binomial[(n-5)/8,(n-5)/24],17,24Binomial[(n-9)/8,(n-17)/24],,0]/(n+7),0]],{n,50}] (* _Robert A. Russell, Mar 29 2024 *)

a(n) = (1/2)*(A047749(n) - 2*A047773(n) - 2*A047760(n) - A047753(n) - A047751(n) - A047764(n) - A047765(n)).

G.f.: (2 - G(z^4) - G(z^6))/z + (G(z^2) + z*G(z^2)^2 - G(z^4) + z*G(z^4) - z^2*G(z^4)^2 + z^2*G(z^6) + z^2*G(z^12) + z^8*G(z^12)^2) / 2 + z - z*G(z^4)^2 - z*G(z^6) - z^2*G(z^6)^2 - z^4*G(z^6)^2 + z^5*G(z^24) + z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 29 2024

A369315 Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.

Original entry on oeis.org

2, 9, 48, 231, 1188, 6114, 32448, 175032, 962472, 5370524, 30377504, 173816313, 1004823816, 5861490300, 34468767840, 204161269620, 1217143807770, 7299003615537, 44005594027200, 266608363362900

Offset: 4

Robert A. Russell, Jan 19 2024

A stereographic projection of the {4,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

			 __ __ __    __ __ __     __ __          __ __
|__|__|__|  |__|__|__|   |__|__|__    __|__|__|  a(4) = 2.
      |__|  |__|            |__|__|  |__|__|

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A369314 {3,oo}.

p=4; Table[(Binomial[(p-1)n,n]/(((p-2)n+1)((p-2)n+2))-If[OddQ[n],If[OddQ[p],Binomial[(p-1)n/2,(n-1)/2]/n,(p+1)Binomial[((p-1)n-1)/2,(n-1)/2]/((p-2)n+2)-Binomial[((p-1)n+1)/2,(n-1)/2]/((p-1)n+1)],Binomial[(p-1)n/2,n/2]/((p-2)n+2)]+DivisorSum[GCD[p,n-1],EulerPhi[#]Binomial[((p-1)n+1)/#,(n-1)/#]/((p-1)n+1)&,#>1&])/2,{n,4,30}]
Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],6Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],2Binomial[3n/2,n/2]]/(n+1))/8,{n,0,30}] (* Robert A. Russell, Jun 19 2025 *)

a(n) = A005034(n) - A005036(n) = (A005034(n) - A047749(n)) / 2 = A005036(n) - A047749(n).

G.f.: (3*G(z) - G(z)^2 - 2*G(z^2) - 3z*G(z^2)^2 + 2z*G(z^4)) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025

cp's OEIS Frontend

A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3.

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A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

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A047765 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type P.

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A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.

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A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

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A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

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A047762 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type E.

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A047774 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type C.

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