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Number of (undirected) paths in the ladder graph P_2 X P_n.
5

%I #15 Jun 30 2017 10:01:13

%S 1,12,49,146,373,872,1929,4118,8589,17644,35889,72538,146021,293200,

%T 587801,1177278,2356541,4715412,9433537,18870210,37744021,75492152,

%U 150988969,301983206,603972333,1207951292,2415909969,4831828138,9663665349,19327340704

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

%H Andrew Howroyd, <a href="/A288516/b288516.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).

%F a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3. - _Eric W. Weisstein_, Jun 30 2017

%F 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.

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

%F a(n) = 18*(2^n-1) - (41*n)/3 - 3*n^2 - n^3/3. - _Colin Barker_, Jun 11 2017

%t Table[18 (2^n - 1) - n (n^2 + 9 n + 41)/3, {n, 20}] (* _Eric W. Weisstein_, Jun 30 2017 *)

%t LinearRecurrence[{6, -14, 16, -9, 2}, {1, 12, 49, 146, 373}, 20] (* _Eric W. Weisstein_, Jun 30 2017 *)

%t 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 *)

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

%o (PARI) a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3 \\ _Charles R Greathouse IV_, Jun 30 2017

%Y Row 2 of A288518.

%Y Cf. A288032, A137882, A287992, A020874.

%K nonn,easy

%O 1,2

%A _Andrew Howroyd_, Jun 10 2017