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%I #27 Apr 09 2023 11:50:02
%S 3,4,-17,-24,99,140,-577,-816,3363,4756,-19601,-27720,114243,161564,
%T -665857,-941664,3880899,5488420,-22619537,-31988856,131836323,
%U 186444716,-768398401,-1086679440,4478554083,6333631924,-26102926097,-36915112104,152139002499,215157040700
%N Expansion of g.f. (1+x)*(3+x)/(1+6*x^2+x^4).
%C From _Creighton Dement_, Dec 18 2004: (Start)
%C Define the following sequences:
%C b(2n) = c(2n+1), b(2n+1) = c(2n); (c(n)) = (1, -3, -7, 17, 41, -99, -239, 577, 1393, -3363, -8119, 19601, 47321). This is the sequence A001333, apart from signs. Then c(2n) = ((-1)^n)*A002315(n) and c(2n+1) = ((-1)^(n+1))*A001541(n+1).
%C (d(n)) = (2, 4, -10, -24, 58, 140, -338, -816, 1970, 4756, -11482, -27720). This is A052542, apart from signs. Also, d(2n) = ((-1)^n)*A075870(n), d(2n+1) = ((-1)^n)*A005319(n+1).
%C (e(n)) = (1, -1, -5, 5, 29, -29, -169, 169, 985, -985, -5741, 5741, 33461, -33461), e(2n) = d(2n)/2, e(2n+1) = - d(2n)/2.
%C (f(n)) = (2, 2, -12, -12, 70, 70, -408, -408, 2378, 2378, -13860, -13860, ) f(2n) = f(2n+1) = d(2n+1)/2.
%C (g(n)) = (0, -3, 0, 17, 0, -99, 0, 577, 0, -3363, 0, 19601, 0, -114243, 0, 665857), g(2n) = 0, g(2n+1) = c(2n+1).
%C Then a(2n) = - c(2n+1), a(2n+1) = d(2n+1) and we have the following conjectures: c(n) + d(n) = e(n) + f(n) = g(n) + a(n); c(n) + d(n) = b(n). In other words, the sequences (c(n) + d(n)) = (e(n) + f(n)) = (g(n) + h(n)) all represent the sequence c with even- and odd-indexed terms reversed. (End)
%H G. C. Greubel, <a href="/A100434/b100434.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-6,0,-1).
%F a(n) = (-1)^floor(n/2)*A000034(n)*A126354(n+3). - _R. J. Mathar_, Mar 08 2009
%F a(n) = -2*a(n-1) - 3*a(n-2) if n is even; a(n) = (4*a(n-1) - a(n-2))/3 if n is odd. - _R. J. Mathar_, Jun 18 2014
%t LinearRecurrence[{0,-6,0,-1}, {3,4,-17,-24}, 41] (* _G. C. Greubel_, Apr 09 2023 *)
%o (Magma) I:=[3,4,-17,-24]; [n le 4 select I[n] else -6*Self(n-2)-Self(n-4): n in [1..40]]; // _G. C. Greubel_, Apr 09 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A100434
%o if (n<4): return (3,4,-17,-24)[n]
%o else: return -6*a(n-2) - a(n-4)
%o [a(n) for n in range(41)] # _G. C. Greubel_, Apr 09 2023
%Y Bisections give A001541, A005319.
%Y Cf. A000034, A001333, A002315, A052542, A075870, A126354.
%K sign,easy
%O 0,1
%A _N. J. A. Sloane_, Nov 21 2004, suggested by correspondence from _Creighton Dement_
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