OFFSET
1,1
COMMENTS
For n > 2, also the number of connected (non-null) induced subgraphs in the n-barbell graph.
LINKS
Jason Bard, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).
FORMULA
a(n) = 2^(n + 1) + 4^(n - 1) - 2.
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: x*(3 - 11*x + 2*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)).
E.g.f.: (exp(4*x) + 8*exp(2*x) - 8*exp(x) - 1)/4. - Stefano Spezia, Apr 26 2025
MATHEMATICA
Table[2^(n + 1) + 4^(n - 1) - 2, {n, 20}]
LinearRecurrence[{7, -14, 8}, {3, 10, 30}, 40]
CoefficientList[Series[(-3 + 11 x - 2 x^2)/(-1 + 7 x - 14 x^2 + 8 x^3), {x, 0, 20}], x]
PROG
(PARI) a(n)=2^(n+1)+4^(n-1)-2 \\ Charles R Greathouse IV, Aug 09 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 09 2017
STATUS
approved