A087455 Expansion of (1 - x)/(1 - 2*x + 3*x^2) in powers of x.

1, 1, -1, -5, -7, 1, 23, 43, 17, -95, -241, -197, 329, 1249, 1511, -725, -5983, -9791, -1633, 26107, 57113, 35905, -99529, -306773, -314959, 290401, 1525679, 2180155, -216727, -6973919, -13297657, -5673557, 28545857, 74112385, 62587199, -97162757, -382087111, -472685951

Offset: 0

Author

Simone Severini, Oct 23 2003

Keywords

• easy
• sign

Comments

Type 2 generalized Gaussian Fibonacci integers.

Binomial transform of A077966. - Philippe Deléham, Dec 02 2008

The real component of Q^n, where Q is the quaternion 1 + 0*i + 1*j + 1*k. - Stanislav Sykora, Jun 11 2012

If entries are multiplied by 2*(-1)^n, which gives 2, -2, -2, 10, -14, -2, 46, -86, 34, 190, -482, 394, ..., we obtain the Lucas V(-2,3) sequence. - R. J. Mathar, Jan 08 2013

The real component of (1 + sqrt(-2))^n. - Giovanni Resta, Apr 01 2014

It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017

Given an alternated cubic honeycomb with a planar dissection along a plane from edge to opposite edge of the containing cube. The sequence (1 + sqrt(-2))^n contains a real component representing distance along the edge of the tetrahedron/octahedron and an imaginary component representing the orthogonal distance along the sqrt(2) axis in a tetrahedron/octahedron, this generates a unique cevian (line from the apical vertex to a vertex on the triangular tiling composing the opposite face) in this plane with length (sqrt(3))^n. - Jason Pruski, Sep 04 2017, Jan 08 2018

From Peter Bala, Apr 01 2018: (Start)

This sequence is the Lucas sequence V(n,2,3). The companion Lucas sequence U(n,2,3) is A088137.

Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 1 o 1 o ... o 1 (n terms) = A088137(n)/a(n). Cf. A025172 and A127357. (End)

Examples

			G.f. = 1 + x - x^2 - 5*x^3 - 7*x^4 + x^5 + 23*x6 + 43*x^7 + 17*x^8 - 95*x^9 + ...
		

References

• Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
• S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).

Links

• Robert Israel, Table of n, a(n) for n = 0..3500
• Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
• A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
• F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
• M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
• Wikipedia, Lucas sequence
• Index entries for linear recurrences with constant coefficients, signature (2,-3).
• Index entries for sequences related to Benford's law

Crossrefs

Cf. A025172, A048473, A077966, A084102, A088137, A088138, A098158, A124182, A127357.

Programs

Magma

[n le 2 select 1 else 2*Self(n-1) -3*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 03 2024

Maple

Digits:=100; a:=n->round(abs(evalf((3^(n/2))*cos(n*arctan(sqrt(2))))));
# alternative:
a:= gfun:-rectoproc({a(n) = 2*a(n-1) - 3*a(n-2),a(0)=1,a(1)=1},a(n),remember):
map(a, [$0..100]); # Robert Israel, Jun 23 2015

Mathematica

CoefficientList[Series[(1-x)/(1-2*x+3*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Apr 01 2014 *)
a[ n_] := ChebyshevT[ n, 1/Sqrt[3]] Sqrt[3]^n // Simplify; (* Michael Somos, May 15 2015 *)
LinearRecurrence[{2,-3},{1,1},50] (* Harvey P. Dale, Jul 30 2019 *)

PARI

{a(n) = real( (1 + quadgen(-8))^n )}; /* Michael Somos, Jul 26 2006 */

PARI

{a(n) = real( subst( poltchebi(n), 'x, quadgen(12) / 3) * quadgen(12)^n)}; /* Michael Somos, Jul 26 2006 */

PARI

a(n)=simplify(polchebyshev(n,,quadgen(12)/3)*quadgen(12)^n) \\ Charles R Greathouse IV, Jun 26 2013

SageMath

[sqrt(3)^n*chebyshev_T(n, 1/sqrt(3)) for n in range(41)] # G. C. Greubel, Jan 03 2024

Formula

a(n) = (3^(n/2))*cos(n*arctan(sqrt(2))). - Paul Barry, Oct 23 2003

From Paul Barry, Sep 03 2004: (Start)

a(n) = 2*a(n-1) - 3*a(n-2).

a(n) = (-1)^n*Sum_{m=0..n} binomial(n, m)*Sum_{k=0..n} binomial(m, 2k)2^(m-k).

Binomial transform of 1/(1 + 2*x^2), or (1, 0, -2, 0, 4, 0, -8, 0, 16, ...). (End)

a(n+1) = a(n+2) - 2*A088137(n+1), a(n+1) = A088137(n+2) - A088137(n+1). - Creighton Dement, Oct 28 2004

a(n) = upper left and lower right terms of [1,-2, 1,1]^n. - Gary W. Adamson, Mar 28 2008

a(n) = Sum_{k=0..n} A098158(n,k)*(-2)^(n-k). - Philippe Deléham, Nov 14 2008

a(n) = Sum_{k=0..n} A124182(n,k)*(-3)^(n-k). - Philippe Deléham, Nov 15 2008

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

a(n) = a(-n) * 3^n for all n in Z. - Michael Somos, Aug 25 2014

E.g.f.: (1/2)*(exp((1 - i*sqrt(2))*x) + exp((1 + i*sqrt(2))*x)), where i is the imaginary unit. - Stefano Spezia, Jul 17 2019

Extensions

The explicit formula was given by Paul Barry.

Corrected and extended by N. J. A. Sloane, Aug 01 2004

More terms from Creighton Dement, Jul 31 2004