A003453 -id:A003453

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A005744

Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)).
(Formerly M3360)

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Milan Janjic, 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).

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Additional comments from Alford Arnold

STATUS

approved

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.

+10
8

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,8

LINKS

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

EXAMPLE

Triangle begins: (n >= 3, k >= 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;

...

PROG

(PARI) \\ 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)

CROSSREFS

Row sums are A001004.

Column k=3 is A003453.

Diagonals include A000207, A003449, A003450.

Cf. A295260, A295419, A295633.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Nov 24 2017

STATUS

approved

A003451

Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.
(Formerly M3330)

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

5,2

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=5..52.

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).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(PARI) \\ 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

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

CROSSREFS

Column 3 of A295633.

Cf. A003453, A006584, A027656.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,18

COMMENTS

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)

Wikipedia, Cactus graph.

Index entries for sequences related to cacti.

FORMULA

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

EXAMPLE

Triangle begins:

0, 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;

...

PROG

(PARI)

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]))}

CROSSREFS

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).

KEYWORD

nonn,tabf,new

AUTHOR

Andrew Howroyd, Feb 24 2025

STATUS

approved

A295622

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

5,1

LINKS

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.

FORMULA

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

PROG

(PARI) \\ See A003442 for DissectionsModCyclicRooted()

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

CROSSREFS

Cf. A003442, A003451, A003452, A003453.

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Nov 24 2017

STATUS

approved

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