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%I #38 Dec 23 2022 07:38:55
%S 2,8,16,28,44,64,88,116,148,184,224,268,316,368,424,484,548,616,688,
%T 764,844,928,1016,1108,1204,1304,1408,1516,1628,1744,1864,1988,2116,
%U 2248,2384,2524,2668,2816,2968,3124,3284,3448,3616,3788,3964,4144,4328,4516,4708,4904,5104,5308,5516,5728,5944,6164,6388,6616,6848,7084,7324,7568,7816
%N Number of (directed) Hamiltonian paths in the n-ladder graph.
%H G. C. Greubel, <a href="/A137882/b137882.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian 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_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F For n>2, m=p^3*q (p,q = primes), a(n) = Sum_{d|m}, (n-1)^(bigomega(d)-omega(d)) = Sum_{d|m}, (n-1)^(A001222(d)-A001221(d))). - _Jaroslav Krizek_, Sep 24 2009
%F For n>1, a(n) = 2*(n^2-n+2); first diagonal of [A154685]. - _Vincenzo Librandi_, Nov 24 2010
%F G.f.: 2*x*(1+x-x^2+x^3)/(1-x)^3. - _Colin Barker_, Jan 20 2012
%F Sum_{n>=1} 1/a(n) = 1/4 + Pi*tanh(sqrt(7)*Pi/2)/(2*sqrt(7)). - _Amiram Eldar_, Dec 23 2022
%p A137882:=n->2*(n^2-n+2): 2,seq(A137882(n), n=2..100); # _Wesley Ivan Hurt_, Apr 25 2017
%t CoefficientList[Series[2*x*(1 + x - x^2 + x^3)/(1 - x)^3, {x,0,50}], x] (* _G. C. Greubel_, Apr 25 2017 *)
%t LinearRecurrence[{3,-3,1},{2,8,16,28},70] (* _Harvey P. Dale_, Nov 15 2018 *)
%o (PARI) my(x='x+O('x^50)); Vec(2*x*(1 + x - x^2 + x^3)/(1 - x)^3) \\ _G. C. Greubel_, Apr 25 2017
%Y Cf. A001221, A001222, A154685.
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, Feb 20 2008
%E Extended and formula corrected by _Max Alekseyev_, Apr 11 2009
%E Corrected the formula which was confusing offsets - _R. J. Mathar_, Jun 04 2010
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