%I #46 Jun 06 2025 14:47:00

%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="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://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

%F From _Elmo R. Oliveira_, Jun 06 2025: (Start)

%F E.g.f.: 2*(exp(x)*(2 + x^2) - (2 + x)).

%F a(n) = 2*A003682(n).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)

%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, A003682, 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