A288516 Number of (undirected) paths in the ladder graph P_2 X P_n.

1, 12, 49, 146, 373, 872, 1929, 4118, 8589, 17644, 35889, 72538, 146021, 293200, 587801, 1177278, 2356541, 4715412, 9433537, 18870210, 37744021, 75492152, 150988969, 301983206, 603972333, 1207951292, 2415909969, 4831828138, 9663665349, 19327340704

Offset: 1

Author

Andrew Howroyd, Jun 10 2017

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Graph Path
• Eric Weisstein's World of Mathematics, Ladder Graph
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Crossrefs

Row 2 of A288518.

Cf. A288032, A137882, A287992, A020874.

Programs

Mathematica

Table[18 (2^n - 1) - n (n^2 + 9 n + 41)/3, {n, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
LinearRecurrence[{6, -14, 16, -9, 2}, {1, 12, 49, 146, 373}, 20] (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(-1 - 6 x + 9 x^2 - 4 x^3)/((-1 + x)^4 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

PARI

Vec((1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x))+O(x^25))

PARI

a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3 \\ Charles R Greathouse IV, Jun 30 2017

Formula

a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3. - Eric W. Weisstein, Jun 30 2017

a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n > 5.

G.f.: x*(1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x)).

a(n) = 18*(2^n-1) - (41*n)/3 - 3*n^2 - n^3/3. - Colin Barker, Jun 11 2017