cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A269869 Number of matchings (not necessarily perfect) in the triangle graph of order n.

Original entry on oeis.org

1, 4, 27, 425, 14278, 1054396, 169858667, 59811185171, 46012925161519, 77344464552678876, 284066030784415134855, 2279568155737623235728996, 39969481180418160836567285156, 1531253921482570179838977438893104, 128176575381689893022287259560629125869
Offset: 1

Views

Author

Andrew Howroyd, Mar 06 2016

Keywords

Comments

The triangle graph of order n has n rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(n) vertices and 3*A000217(n-1) edges altogether.

References

  • Samuel Dittmer, Hiram Golze, Grant Molnar, and Caleb Stanford, Puzzle and Proof: A Decade of Problems from the Utah Math Olympiad, CRC Press, 2025, p. 59.

Links

Crossrefs

Cf. A000217, A028420, A039907, A112676, A178446.
Row sums of A288852.

A039907 Number of perfect matchings in triangle graph with n nodes per side.

Original entry on oeis.org

1, 0, 0, 2, 6, 0, 0, 2196, 37004, 0, 0, 2317631400, 216893681800, 0, 0, 2326335506123418128, 1208982377794384163088, 0, 0, 2220650888749669503773432361504, 6408743336016148761893699822360672, 0, 0, 2015895925780490675949731718780144934779733312
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 17).

Links

Crossrefs

Cf. A071093, A269869, A178446.

Programs

  • Maple
    with(LinearAlgebra): a:= proc(n) option remember; local l, ll, i, j, h0, h1, M; if n=0 then return 1 fi; if n<0 or member(irem(n, 4), [1, 2]) then return 0 fi; l:= []; for j from 1 to n-1 do h0:= j*(j-1)/2+1; h1:= j*(j+1)/2+1; for i from 1 to j do l:= [l[], [h1, h1+1]]; if irem(i, 2)=1 then l:= [l[], [h1, h0]]; h1:= h1+1; l:=[l[], [h1, h0]]; h0:=h0+1 else l:= [l[], [h0, h1]]; h1:= h1+1; l:=[l[], [h0, h1]]; h0:=h0+1 fi od od; M:= Matrix((n+1)*n/2); for ll in l do M[ll[1], ll[2]]:= 1; M[ll[2], ll[1]]:= -1 od: isqrt(Determinant(M)) end: seq(a(n), n=0..20); # Alois P. Heinz, May 08 2010
  • Mathematica
    a[n_] := a[n] = Module[{l, ll, i, j, h0, h1, M}, If[n == 0 , Return[1]]; If[n < 0 || MemberQ[{1, 2}, Mod[n, 4]], Return[0]]; l = {}; For[j = 1, j <= n-1, j++, h0 = j*(j-1)/2+1; h1 = j*(j+1)/2+1; For[i = 1, i <= j, i++, l = Join[l, {h1, h1+1}]; If[Mod [i, 2] == 1, l = Join[l, {h1, h0}]; h1 = h1+1; l = Join[l, {h1, h0}]; h0 = h0+1, l = Join[l, {h0, h1}]; h1 = h1+1; l = Join[l, {h0, h1}]; h0 = h0+1]]]; M[, ] = 0; Do[M[ll[[1]], ll[[2]]] = 1; M[ll[[2]], ll[[1]]] = -1, {ll, Partition[l, 2]}]; Sqrt[Det[Array[M, {n*(n+1)/2, n*(n+1)/2}]]]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *)

Formula

a(n) = A178446(n) if A000217(n) is even, a(n) = 0 otherwise. - Andrew Howroyd, Mar 06 2016

Extensions

a(17)-a(20) from Alois P. Heinz, May 08 2010
a(21)-a(23) from Alois P. Heinz, Jan 12 2014

A071093 Number of perfect matchings in triangle graph with n nodes per side as n runs through numbers congruent to 0 or 3 mod 4.

Original entry on oeis.org

1, 2, 6, 2196, 37004, 2317631400, 216893681800, 2326335506123418128, 1208982377794384163088, 2220650888749669503773432361504, 6408743336016148761893699822360672, 2015895925780490675949731718780144934779733312, 32307672245407537492814937397129549558917000333504
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 17).

Links

Crossrefs

Cf. A039907, A178446, A269869.

Programs

  • Maple
    with(LinearAlgebra): b:= proc(n) option remember; local l, ll, i, j, h0, h1, M; if n=0 then return 1 fi; if n<0 or member(irem(n, 4), [1, 2]) then return 0 fi; l:= []; for j from 1 to n-1 do h0:= j*(j-1)/2+1; h1:= j*(j+1)/2+1; for i from 1 to j do l:= [l[], [h1, h1+1]]; if irem(i, 2)=1 then l:= [l[], [h1, h0]]; h1:= h1+1; l:=[l[], [h1, h0]]; h0:=h0+1 else l:= [l[], [h0, h1]]; h1:= h1+1; l:=[l[], [h0, h1]]; h0:=h0+1 fi od od; M:= Matrix((n+1)*n/2); for ll in l do M[ll[1], ll[2]]:= 1; M[ll[2], ll[1]]:= -1 od: isqrt(Determinant(M)); end: a:= n-> b(2*n +irem(n, 2)): seq(a(n), n=0..10); # Alois P. Heinz, May 08 2010
  • Mathematica
    b[n_] := b[n] = Module[{l, ll, i, j, h0, h1, M}, If[n == 0 , Return[1]]; If[n<0 || MatchQ[Mod[n, 4], 1|2] , Return[0]]; l = {}; For[j = 1, j <= n-1, j++, h0 = j*(j-1)/2+1; h1 = j*(j+1)/2+1; For[i = 1, i <= j, i++, AppendTo[l, {h1, h1+1}]; If[Mod[i, 2] == 1, AppendTo[l, {h1, h0}]; h1++; AppendTo[l, {h1, h0}]; h0++ , AppendTo[l, {h0, h1}]; h1++; AppendTo[l, {h0, h1}]; h0++ ]]]; M[, ] = 0; (M[#[[1]], #[[2]]] = 1; M[#[[2]], #[[1]]] = -1)& /@ l; Sqrt[Det[Array[M, {n*(n+1)/2, n*(n+1)/2}]]]]; a[n_] := b[2*n + Mod[n, 2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 29 2014, after Alois P. Heinz *)

Formula

a(2n) = A039907(4n) = A178446(4n), a(2n+1) = A039907(4n+3) = A178446(4n+3). - Andrew Howroyd, Mar 06 2016

Extensions

a(9)-a(10) from Alois P. Heinz, May 08 2010
a(11)-a(12) from Alois P. Heinz, Jan 12 2014
Showing 1-3 of 3 results.

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