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A113249 Corresponds to m = 3 in a family of 4th-order linear recurrence sequences given by a(m,n) = m^4*a(n-4) + (2*m)^2*a(n-3) - 4*a(n-1), a(m,0) = -1, a(m,1) = 4, a(m,2) = -13 + 6*(m-1) + 3*(m-1)^2, a(m,3) = (-8+m^2)^2.

%I A113249 #29 Aug 18 2023 23:51:23
%S A113249 -1,4,11,1,59,484,-1009,6241,-2761,13924,87251,57121,49139,4072324,
%T A113249 -7165609,35058241,10350959,30492484,559712411,973502401,-1957852501,
%U A113249 30450948004,-41421000289,174055005601,241428053159,9658565284,2872244917091,11300885699041,-25300162140061
%N A113249 Corresponds to m = 3 in a family of 4th-order linear recurrence sequences given by a(m,n) = m^4*a(n-4) + (2*m)^2*a(n-3) - 4*a(n-1), a(m,0) = -1, a(m,1) = 4, a(m,2) = -13 + 6*(m-1) + 3*(m-1)^2, a(m,3) = (-8+m^2)^2.
%C A113249 Conjecture: a(m, 2*n+1) is a perfect square for all m, n. Disregarding signs and/or initial term, we have: m = 0 (A000302), m = 1 (A097948), m = 2 (A056450), m = 3 (a(n)), m = 4 (A113250), m = 5 (A113251), m = 6 (A113252), m = 7 (A113253), m = 8 (A113254), m = 9 (A113255), m = 10 (A113256).
%C A113249 In this case, a(2n+1) = b(n)^2 where b(n) = Re((2+sqrt(-5))^(n+1)) satisfies the recurrence b(n) = 4*b(n-1) - 9*b(n-2) with b(0)=2, b(1)=-1. - _Robert Israel_, Oct 23 2017
%H A113249 Colin Barker, <a href="/A113249/b113249.txt">Table of n, a(n) for n = 0..1000</a>
%H A113249 Robert Munafo, <a href="http://www.mrob.com/pub/math/seq-floretion.html">Sequences Related to Floretions</a>
%H A113249 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,0,36,81).
%F A113249 G.f.: (-1+27*x^2+81*x^3)/((-3*x+1)*(3*x+1)*(9*x^2+4*x+1)).
%F A113249 a(2k+1) = (2*A176333(k)-3*A190967(k))^2. - _Robert Israel_, Oct 23 2017
%F A113249 a(n) = -4*a(n-1) + 36*a(n-3) + 81*a(n-4) for n>3. - _Colin Barker_, May 19 2019
%F A113249 a(n) = 3^(n+1)*(1 - (-1)^n + 2*cos(arccos(-2/3)*(n+1)))/4. - _Eric Simon Jacob_, Aug 06 2023
%e A113249 a(3, 13) = 93161710957356599364/((-2+i*sqrt(5))^14*(2+i*sqrt(5))^14) = 4072324 = 2^2*1009^2.
%p A113249 f:= gfun:-rectoproc({a(n) = 81*a(n-4)+36*a(n-3)-4*a(n-1),a(0) = -1, a(1) = 4, a(2) = 11, a(3) = 1},a(n),remember):
%p A113249 map(f, [$0..30]); # _Robert Israel_, Oct 23 2017
%t A113249 LinearRecurrence[{-4, 0, 36, 81}, {-1, 4, 11, 1}, 29] (* _Jean-François Alcover_, Sep 25 2017 *)
%o A113249 (PARI) Vec(-(1 - 27*x^2 - 81*x^3) / ((1 - 3*x)*(1 + 3*x)*(1 + 4*x + 9*x^2)) + O(x^30)) \\ _Colin Barker_, May 19 2019
%Y A113249 Cf. A000302, A097948, A056450, A113250, A113251, A113252, A113253, A113254, A113255, A113256.
%Y A113249 Cf. A176333, A190967.
%K A113249 easy,sign
%O A113249 0,2
%A A113249 _Creighton Dement_, Oct 20 2005