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A137882 Number of (directed) Hamiltonian paths in the n-ladder graph. 11
2, 8, 16, 28, 44, 64, 88, 116, 148, 184, 224, 268, 316, 368, 424, 484, 548, 616, 688, 764, 844, 928, 1016, 1108, 1204, 1304, 1408, 1516, 1628, 1744, 1864, 1988, 2116, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Ladder Graph.
FORMULA
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
For n>1, a(n) = 2*(n^2-n+2); first diagonal of [A154685]. - Vincenzo Librandi, Nov 24 2010
G.f.: 2*x*(1+x-x^2+x^3)/(1-x)^3. - Colin Barker, Jan 20 2012
Sum_{n>=1} 1/a(n) = 1/4 + Pi*tanh(sqrt(7)*Pi/2)/(2*sqrt(7)). - Amiram Eldar, Dec 23 2022
MAPLE
A137882:=n->2*(n^2-n+2): 2, seq(A137882(n), n=2..100); # Wesley Ivan Hurt, Apr 25 2017
MATHEMATICA
CoefficientList[Series[2*x*(1 + x - x^2 + x^3)/(1 - x)^3, {x, 0, 50}], x] (* G. C. Greubel, Apr 25 2017 *)
LinearRecurrence[{3, -3, 1}, {2, 8, 16, 28}, 70] (* Harvey P. Dale, Nov 15 2018 *)
PROG
(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
CROSSREFS
Sequence in context: A322945 A157512 A252594 * A194643 A327329 A136514
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Feb 20 2008
EXTENSIONS
Extended and formula corrected by Max Alekseyev, Apr 11 2009
Corrected the formula which was confusing offsets - R. J. Mathar, Jun 04 2010
STATUS
approved

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Last modified June 12 18:11 EDT 2024. Contains 373359 sequences. (Running on oeis4.)