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%I #13 Jun 17 2017 03:04:39
%S 26,102,178,254,330,406,482,558,634,710,786,862,938,1014,1090,1166,
%T 1242,1318,1394,1470,1546,1622,1698,1774,1850,1926,2002,2078,2154,
%U 2230,2306,2382,2458,2534,2610,2686,2762,2838,2914,2990,3066,3142,3218,3294,3370
%N Linear bound on the genera of Heegaard splittings of closed, orientable 3-manifolds that admit a generalized triangulation with n generalized tetrahedra.
%C Let N be a closed, orientable 3-manifold that admits a triangulation with t tetrahedra. Let F be a Heegaard surface for N. S. Schleimer showed that if g(F) >= 2^{2^{16}t^2}, then the Hempel distance of F (denoted by d(F)) is at most two. In this paper we prove the following generalization:
%C Let M be an orientable 3-manifold that admits a generalized triangulation with t generalized tetrahedra. Let S be a Heegaard surface for M. If g(S) >= 76t+26, then d(S) <= 2.
%H Tsuyoshi Kobayashi, Yo'av Rieck, <a href="http://arXiv.org/abs/0803.2751">A linear bound on the genera of Heegaard splittings with distances greater than two</a>, arXiov:0803.2751 March 20, 2008.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 76*n + 26.
%F a(n) = 2*a(n-1)-a(n-2). G.f.: 2*(25*x+13)/(x-1)^2. [_Colin Barker_, Nov 09 2012]
%t LinearRecurrence[{2,-1},{26,102},50] (* _Harvey P. Dale_, Oct 14 2013 *)
%K easy,nonn
%O 0,1
%A _Jonathan Vos Post_, Mar 20 2008
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