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%I #17 Mar 14 2024 13:54:24
%S 1,1,-2,-4,0,16,0,0,-128,256,-512,2048,-6144,16384,-49152,147456,
%T -425984,1245184,-3670016,10747904,-31457280,92274688,-270532608,
%U 792723456,-2323644416,6811549696,-19964887040,58518929408,-171530256384,502779609088,-1473710653440,4319663357952
%N a(n) = - 2*a(n-1) - 8*a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.
%C Were only the first few terms of the sequence taken into account, it would appear that the sequence consists only of 0 and powers of 2. Upon dividing out the powers of 2 from each term of the sequence, one sees that, actually, a(n) = c(n)*d(n) with c = (1, 1, -1, -1, 0, 1, 0, 0, 1, 1, 1, 1, 3, 1, 3, 3, 13, 19, 7, 41, 15, 11, 129, 189, ...) and d = (2^0, 2^0, 2, 2^2, 0, 2^4, 0, 0, 2^7, 2^8, 2^9, 2^11, 2^11, 2^14, 2^14, 2^14, 2^15, 2^16, 2^19, 2^18, 2^21, 2^23, 2^21, 2^22, ...).
%C Floretion Algebra Multiplication Program, FAMP Code: 1basej[A*B] with A = + .5'i - .5'k + .5i' - .5k' - 2'jj' - .5'ij' - .5'ji' - .5'jk' - .5'kj' and B = + .5'i + .5'j + .5i' + .5j' + .5'kk' + .5'ij' + .5'ji' + .5e
%H G. C. Greubel, <a href="/A106603/b106603.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,-8).
%F G.f.: (1 + 3*x)/(1 + 2*x + 8*x^3).
%F a(n) = b(n) + 3*b(n-1), where b(n) = 2^n*A199804(n). - _R. J. Mathar_, Sep 11 2019
%F a(n) = (-2)^(n-1)*(-2*c(n) + 3*c(n-1)), where c(n) = A000930(n). - _G. C. Greubel_, Sep 08 2021
%t LinearRecurrence[{-2,0,-8}, {1,1,-2}, 41] (* _G. C. Greubel_, Sep 08 2021 *)
%o (Magma) I:=[1,1,-2]; [n le 3 select I[n] else (-2)*(Self(n-1) +4*Self(n-3)): n in [1..41]]; // _G. C. Greubel_, Sep 08 2021
%o (SageMath)
%o def A106603_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+3*x)/(1+2*x+8*x^3) ).list()
%o A106603_list(40) # _G. C. Greubel_, Sep 08 2021
%Y Cf. A000930, A199804.
%K easy,sign
%O 0,3
%A _Creighton Dement_, May 10 2005
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