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A157456 Expansion of x * (1 - x) / (1 - 16*x + x^2).

%I A157456 #50 May 04 2025 09:07:52
%S A157456 1,15,239,3809,60705,967471,15418831,245733825,3916322369,62415424079,
%T A157456 994730462895,15853271982241,252657621252961,4026668668065135,
%U A157456 64174041067789199,1022757988416562049,16299953773597203585,259776502389138695311,4140124084452621921391
%N A157456 Expansion of x * (1 - x) / (1 - 16*x + x^2).
%C A157456 Positive values of x (or y) satisfying x^2 - 16xy + y^2 + 14 = 0. - _Colin Barker_, Feb 11 2014
%H A157456 Vincenzo Librandi, <a href="/A157456/b157456.txt">Table of n, a(n) for n = 1..200</a>
%H A157456 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H A157456 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A157456 G.f.: x*(1-x) / ( 1-16*x+x^2 ). - _R. J. Mathar_, Oct 31 2011
%F A157456 a(n) = 16*a(n-1)-a(n-2). - _Colin Barker_, Feb 11 2014
%F A157456 a(n) = (1/18)*(9-sqrt(63))*(1+(8+sqrt(63))^(2*n-1))/(8+sqrt(63))^(n-1). [_Bruno Berselli_, Feb 25 2014]
%F A157456 a(n) = sqrt(2+(8-3*sqrt(7))^(1+2*n)+(8+3*sqrt(7))^(1+2*n))/(3*sqrt(2)). - _Gerry Martens_, Jun 06 2015
%F A157456 a(n) = A077412(n-1) - A077412(n-2). - _R. J. Mathar_, Feb 05 2020
%p A157456 f:= gfun:-rectoproc({a(n)=16*a(n-1)-a(n-2),a(1)=1,a(2)=15},a(n),remember):
%p A157456 map(f, [$1..30]); # _Robert Israel_, Jul 07 2015
%t A157456 CoefficientList[Series[(1 - x)/(1 - 16 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%t A157456 LinearRecurrence[{16,-1},{1,15},20] (* _Harvey P. Dale_, Sep 17 2019 *)
%o A157456 (Magma) I:=[1,15]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 12 2014
%Y A157456 Cf. A159678.
%Y A157456 Cf. similar sequences listed in A238379.
%K A157456 nonn,easy
%O A157456 1,2
%A A157456 _Paul Weisenhorn_, Mar 01 2009
%E A157456 New name (using the g.f. by _R. J. Mathar_) from _Joerg Arndt_, Jun 06 2015