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%I #49 Dec 31 2023 11:27:14
%S 1,5,24,112,512,2304,10240,45056,196608,851968,3670016,15728640,
%T 67108864,285212672,1207959552,5100273664,21474836480,90194313216,
%U 377957122048,1580547964928,6597069766656,27487790694400,114349209288704,474989023199232,1970324836974592,8162774324609024
%N a(0) = 1, a(n) = (n + 4)*4^(n-1) for n >= 1.
%C a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i) = 5, m(i,j) = i/j.
%C Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 3*m(i-1,j-1).
%C 4th binomial transform of (1,1,0,0,0,0,...). - _Paul Barry_, Mar 07 2003
%C Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the complete graph K_n (see A235113). Example: a(1)=5; indeed, K_1 is the one vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}. - _Emeric Deutsch_, Jan 13 2014
%C Row sums of A235113.
%H F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, <a href="http://arxiv.org/abs/math/0702550">A closed formula for the number of convex permutominoes</a>, arXiv:math/0702550 [math.CO], 2007.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).
%F a(n) = 8*a(n-1)-16*a(n-2), a(0) = 1, a(1) = 5. - _Paul Barry_, Mar 07 2003
%F G.f.: (1 - 3*x)/(1 - 4*x)^2. - _Philippe Deléham_, Dec 11 2008
%F From _Amiram Eldar_, Jan 14 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 1024*log(4/3) - 880/3.
%F Sum_{n>=0} (-1)^n/a(n) = 688/3 - 1024*log(5/4). (End)
%F E.g.f.: exp(4*x)*(1 + x). - _Stefano Spezia_, Mar 05 2023
%t LinearRecurrence[{8, -16}, {1, 5}, 22] (* _Jean-François Alcover_, Nov 06 2018 *)
%o (Sage) [lucas_number2(n, 4, 0)*n/2^10 for n in range(4, 26)] # _Zerinvary Lajos_, Mar 13 2009
%Y Cf. A001792, A006234, A081105, A006234, A235113.
%Y Cf. A002697, A034007, A079861, A045891, A087449.
%K nonn,easy
%O 0,2
%A _Benoit Cloitre_, Feb 01 2003
%E More terms from _Stefano Spezia_, Mar 05 2023