A199804 -id:A199804 - cp's OEIS frontend

A199802 G.f.: 1/(1-2x+2x^2-x^3+x^4).

1, 2, 2, 1, -1, -4, -7, -8, -5, 3, 15, 27, 32, 22, -8, -55, -104, -128, -95, 17, 200, 399, 510, 405, -11, -721, -1525, -2024, -1708, -172, 2573, 5806, 8002, 7137, 1503, -9072, -22015, -31520, -29585, -9073, 31519, 83119, 123712, 121778, 47732, -107499, -312396, -483840, -498119, -233455, 357884, 1168399, 1885694, 2025929, 1090985, -1152593

Offset: 0

N. J. A. Sloane, Nov 10 2011

Hirschhorn, Michael D., Non-trivial intertwined second-order recurrence relations, Fibonacci Quart. 43 (2005), no. 4, 316-325. See G_n.
Index entries for linear recurrences with constant coefficients, signature (2,-2,1,-1).

The main sequences mentioned in the Hisrchhorn paper are A199802, A199803, A199744, A199804, A077961, A199805.

CoefficientList[Series[1/(1-2x+2x^2-x^3+x^4),{x,0,60}],x] (* or *) LinearRecurrence[ {2,-2,1,-1},{1,2,2,1},60] (* Harvey P. Dale, May 11 2022 *)

A106603 a(n) = - 2a(n-1) - 8a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.

Original entry on oeis.org

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

Offset: 0

Creighton Dement, May 10 2005

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, ...).

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,0,-8).

Cf. A000930, A199804.

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

LinearRecurrence[{-2,0,-8}, {1,1,-2}, 41] (* G. C. Greubel, Sep 08 2021 *)

def A106603_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+3*x)/(1+2*x+8*x^3) ).list()
A106603_list(40) # G. C. Greubel, Sep 08 2021

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

cp's OEIS Frontend

A199802 G.f.: 1/(1-2x+2x^2-x^3+x^4).

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A106603 a(n) = - 2a(n-1) - 8a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.

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cp's OEIS Frontend

A199802 G.f.: 1/(1-2*x+2*x^2-x^3+x^4).

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Author

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Programs

Mathematica

A106603 a(n) = - 2*a(n-1) - 8*a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A199802 G.f.: 1/(1-2x+2x^2-x^3+x^4).

A106603 a(n) = - 2a(n-1) - 8a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.