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A286814
Number of matchings in the n-helm graph.
0
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
Eric Weisstein's World of Mathematics, Helm Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
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
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 27 2017
STATUS
approved