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

%I A288516 #16 Feb 16 2025 08:33:47
%S A288516 1,12,49,146,373,872,1929,4118,8589,17644,35889,72538,146021,293200,
%T A288516 587801,1177278,2356541,4715412,9433537,18870210,37744021,75492152,
%U A288516 150988969,301983206,603972333,1207951292,2415909969,4831828138,9663665349,19327340704
%N A288516 Number of (undirected) paths in the ladder graph P_2 X P_n.
%H A288516 Andrew Howroyd, <a href="/A288516/b288516.txt">Table of n, a(n) for n = 1..200</a>
%H A288516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H A288516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>
%H A288516 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F A288516 a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3. - _Eric W. Weisstein_, Jun 30 2017
%F A288516 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 A288516 G.f.: x*(1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x)).
%F A288516 a(n) =  18*(2^n-1) - (41*n)/3 - 3*n^2 - n^3/3. - _Colin Barker_, Jun 11 2017
%t A288516 Table[18 (2^n - 1) - n (n^2 + 9 n + 41)/3, {n, 20}] (* _Eric W. Weisstein_, Jun 30 2017 *)
%t A288516 LinearRecurrence[{6, -14, 16, -9, 2}, {1, 12, 49, 146, 373}, 20] (* _Eric W. Weisstein_, Jun 30 2017 *)
%t A288516 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 A288516 (PARI) Vec((1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x))+O(x^25))
%o A288516 (PARI) a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3 \\ _Charles R Greathouse IV_, Jun 30 2017
%Y A288516 Row 2 of A288518.
%Y A288516 Cf. A288032, A137882, A287992, A020874.
%K A288516 nonn,easy
%O A288516 1,2
%A A288516 _Andrew Howroyd_, Jun 10 2017