A003453 -id:A003453 - cp's OEIS frontend

A005744 Expansion of x(1+x-x^2)/((1-x)^4(1+x)).

0, 1, 4, 9, 17, 28, 43, 62, 86, 115, 150, 191, 239, 294, 357, 428, 508, 597, 696, 805, 925, 1056, 1199, 1354, 1522, 1703, 1898, 2107, 2331, 2570, 2825, 3096, 3384, 3689, 4012, 4353, 4713, 5092, 5491, 5910, 6350, 6811, 7294, 7799, 8327, 8878, 9453, 10052

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of n-covers of a 2-set.
Boolean switching functions a(n,s) for s = 2.
Without the initial 0, this is row 1 of the convolution array A213778. - Clark Kimberling, Jun 21 2012
a(n) equals the second column of the triangle A355754. - Eric W. Weisstein, Mar 12 2024

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. V. Jayanthan, S. A. Seyed Fakhari, I. Swanson, and S. Yassemi, Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs, arXiv:2405.06781 [math.AC], 2024. See p. 17.
Vladeta Jovovic, Binary matrices up to row and column permutations.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

John W. Layman observes that A003453 appears to be the alternating sum transform (PSumSIGN) of A005744.
Cf. A002623, A005745, A005746, A005747, A005748, A005771, A003180, A008778, A052265.
Cf. A181971, A213778.
Cf. A355754.

CoefficientList[Series[x (1+x-x^2)/((1-x)^4(1+x)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,4,9,17},50] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,3,-2,-2,3]^n*[0;1;4;9;17])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

a(n) = A002623(n) - (n+1).
a(n) = n*(n-1)/2 + Sum_{j=1..floor((n+1)/2)} (n-2*j+1)*(n-2*j)/2. - N. J. A. Sloane, Nov 28 2003
From R. J. Mathar, Apr 01 2010: (Start)
a(n) = 5*n/12 - 1/16 + 5*n^2/8 + n^3/12 + (-1)^n/16.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)
a(n) = A181971(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) + a(n+1) = A008778(n). - R. J. Mathar, Mar 13 2021
E.g.f.: (x*(2*x^2 + 21*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 27*x - 3)*sinh(x))/24. - Stefano Spezia, Jul 27 2022

Additional comments from Alford Arnold

A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 6, 7, 4, 1, 3, 11, 24, 24, 12, 1, 3, 17, 51, 89, 74, 27, 1, 4, 26, 109, 265, 371, 259, 82, 1, 4, 36, 194, 660, 1291, 1478, 891, 228, 1, 5, 50, 345, 1477, 3891, 6249, 6044, 3176, 733, 1, 5, 65, 550, 3000, 10061, 21524, 29133, 24302, 11326, 2282

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  3,   3;
1, 2,  6,   7,   4;
1, 3, 11,  24,  24,   12;
1, 3, 17,  51,  89,   74,   27;
1, 4, 26, 109, 265,  371,  259,  82;
1, 4, 36, 194, 660, 1291, 1478, 891, 228;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A001004.
Column k=3 is A003453.
Diagonals include A000207, A003449, A003450.
Cf. A295260, A295419, A295633.

\\ See A295419 for DissectionsModDihedral()
T=DissectionsModDihedral(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 4, 8, 16, 25, 40, 56, 80, 105, 140, 176, 224, 273, 336, 400, 480, 561, 660, 760, 880, 1001, 1144, 1288, 1456, 1625, 1820, 2016, 2240, 2465, 2720, 2976, 3264, 3553, 3876, 4200, 4560, 4921, 5320, 5720, 6160, 6601, 7084, 7568, 8096, 8625, 9200, 9776, 10400

Offset: 5

N. J. A. Sloane

nonn

In other words, the number of 2-dissections of an n-gon modulo the cyclic action.

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

D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Column 3 of A295633.

Cf. A003453, A006584, A027656.

[(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015

T51:= proc(n)
if n mod 2 = 0 then n*(n-2)*(n-4)/12;
else (n+1)*(n-3)*(n-4)/12; fi end;
[seq(T51(n),n=5..80)]; # N. J. A. Sloane, Dec 28 2012

Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* Vincenzo Librandi, Apr 05 2015 *)
CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2),{x,0,20}],x] (* Vaclav Kotesovec, Apr 05 2015 *)

Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015

\\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

G.f.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).
See also the Maple code for an explicit formula.
a(n) = A006584(n+3) - A027656(n). - Yosu Yurramendi, Aug 07 2008
a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - Luce ETIENNE, Apr 04 2015

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
First formula adapted to offset by Vaclav Kotesovec, Apr 05 2015
Name clarified by Andrew Howroyd, Nov 25 2017

A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 3, 0, 1, 3, 6, 0, 1, 4, 11, 2, 0, 1, 4, 17, 5, 0, 1, 5, 26, 17, 0, 1, 5, 36, 37, 2, 0, 1, 6, 50, 78, 12, 0, 1, 6, 65, 140, 44, 0, 1, 7, 85, 248, 131, 4, 0, 1, 7, 106, 396, 325, 23

Offset: 0

Andrew Howroyd, Feb 24 2025

All such graphs are cactus graphs (with bridges allowed).

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  3;
  0, 1, 3,  6;
  0, 1, 4, 11,  2;
  0, 1, 4, 17,  5;
  0, 1, 5, 26, 17;
  0, 1, 5, 36, 37, 2;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Columns 0..3 are A000007, A000012(n+3), A004526(n+4), A003453(n+4).
Row sums are A380805.
Cf. A000672, A380631 (with vertices of any degree).

raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
R(n,y)={my(g=O(x^3)); for(n=1, (n-1)\2, my(p=x*(1 + g), p2=raise(p,2)); g=x*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
G(n,y=1)={my(g=R(n,y), p = x*(1+g) + O(x*x^n));
  my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
  my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
  1 + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
{my(A=T(15)); for(i=1, #A, print(A[i]))}

T(3*n,n) = A000672(n).

A295622 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

3, 11, 24, 46, 75, 117, 168, 236, 315, 415, 528, 666, 819, 1001, 1200, 1432, 1683, 1971, 2280, 2630, 3003, 3421, 3864, 4356, 4875, 5447, 6048, 6706, 7395, 8145, 8928, 9776, 10659, 11611, 12600, 13662, 14763, 15941, 17160, 18460, 19803, 21231, 22704, 24266

Offset: 5

Andrew Howroyd, Nov 24 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..500
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Cf. A003442, A003451, A003452, A003453.

\\ See A003442 for DissectionsModCyclicRooted()
{ my(v=DissectionsModCyclicRooted(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) }

Conjectures from Colin Barker, Nov 25 2017: (Start)
G.f.: x^5*(3 + 5*x - x^2 - x^3) / ((1 - x)^4*(1 + x)^2).
a(n) = (n-4)*(-5 + (-1)^n - 4*n + 2*n^2) / 8 for n>4.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>10.
(End)
a(n) = Sum_{k=0..n-5} f(k), where f(n) = Sum_{k=0..n} (3 + lcm(k, 2)) (conjecture). - Jon Maiga, Nov 28 2018

cp's OEIS Frontend

A005744 Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)).

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A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

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A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.

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A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

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A295622 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation.

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A005744 Expansion of x(1+x-x^2)/((1-x)^4(1+x)).