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A106603
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a(n) = - 2*a(n-1) - 8*a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.
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2
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1, 1, -2, -4, 0, 16, 0, 0, -128, 256, -512, 2048, -6144, 16384, -49152, 147456, -425984, 1245184, -3670016, 10747904, -31457280, 92274688, -270532608, 792723456, -2323644416, 6811549696, -19964887040, 58518929408, -171530256384, 502779609088, -1473710653440, 4319663357952
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OFFSET
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0,3
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COMMENTS
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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, ...).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,0,-8).
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FORMULA
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G.f.: (1 + 3*x)/(1 + 2*x + 8*x^3).
a(n) = b(n) + 3*b(n-1), where b(n) = 2^n*A199804(n). - R. J. Mathar, Sep 11 2019
a(n) = (-2)^(n-1)*(-2*c(n) + 3*c(n-1)), where c(n) = A000930(n). - G. C. Greubel, Sep 08 2021
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MATHEMATICA
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LinearRecurrence[{-2, 0, -8}, {1, 1, -2}, 41] (* G. C. Greubel, Sep 08 2021 *)
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PROG
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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
(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
(Sage)
def A106603_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+3*x)/(1+2*x+8*x^3) ).list()
A106603_list(40) # G. C. Greubel, Sep 08 2021
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CROSSREFS
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Cf. A000930, A199804.
Sequence in context: A073451 A078022 A203850 * A059057 A196225 A127511
Adjacent sequences: A106600 A106601 A106602 * A106604 A106605 A106606
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KEYWORD
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easy,sign
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AUTHOR
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Creighton Dement, May 10 2005
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STATUS
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approved
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