A286814 Number of matchings in the n-helm graph.

2, 3, 10, 29, 82, 227, 618, 1661, 4418, 11651, 30506, 79389, 205522, 529635, 1359434, 3476989, 8865026, 22538755, 57157578, 144615709, 365127634, 920110051, 2314564522, 5812911741, 14576950082, 36503608707, 91294323178, 228049363229, 569017421650, 1418290058723

Offset

0,1

Comments

Extended to a(0)-a(2) using the formula.

Links

Table of n, a(n) for n=0..29.

Eric Weisstein's World of Mathematics, Helm Graph

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Formula

a(n) = ((1-sqrt(2))^n*(4-sqrt(2)*n)+(1+sqrt(2))^n*(4+sqrt(2)*n))/4;

a(n) = A002203(n) + n*A000129(n).

a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3)-a(n-4).

G.f.: (2-5*x+2*x^2+3*x^3)/(-1+2*x+x^2)^2.

Mathematica

Table[1/4 ((1 - Sqrt[2])^n (4 - Sqrt[2] n) + (1 + Sqrt[2])^n (4 + Sqrt[2] n)), {n, 0, 20}] // Expand
Table[LucasL[n, 2] + n Fibonacci[n, 2], {n, 0, 20}]
LinearRecurrence[{4, -2, -4, -1}, {3, 10, 29, 82}, {0, 20}]
CoefficientList[Series[(2 - 5 x + 2 x^2 + 3 x^3)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]

Crossrefs

Sequence in context: A034324 A084380 A363139 * A371607 A338583 A121909

Adjacent sequences: A286811 A286812 A286813 * A286815 A286816 A286817

Keyword

nonn,easy

Author

Eric W. Weisstein, May 27 2017

Status

approved