A228842 Binomial transform of A014448.

2, 6, 28, 144, 752, 3936, 20608, 107904, 564992, 2958336, 15490048, 81106944, 424681472, 2223661056, 11643240448, 60964798464, 319215828992, 1671435780096, 8751751364608, 45824765067264, 239941584945152, 1256350449401856, 6578336356630528, 34444616342175744

Offset: 0

Author

R. J. Mathar, Nov 10 2013

Keywords

• nonn
• easy

Comments

The binomial transform of this sequence is 2, 8, 42, 248,... = 2*A108404(n).

References

• C. Smith, A Treatise on Algebra, Macmillan, London, 5th ed., 1950, p. 360, Example 44.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence B_4.
• Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (6,-4).

Crossrefs

Cf. A014448, A108404, A098648.

When divided by 2^n this becomes(essentially) A005248.

Programs

Mathematica

CoefficientList[Series[2*(1 - 3 x)/(1 - 6 x + 4 x^2), {x, 0, 23}], x] (* Michael De Vlieger, Aug 26 2021 *)
LinearRecurrence[{6,-4},{2,6},30] (* Harvey P. Dale, Jun 30 2024 *)

PARI

Vec(2*(1 - 3*x) / (1 - 6*x + 4*x^2) + O(x^30)) \\ Colin Barker, Sep 21 2017

Formula

G.f.: 2*( 1-3*x ) / ( 1-6*x+4*x^2 ).

a(n) = 2*A098648(n).

From Colin Barker, Sep 21 2017: (Start)

a(n) = (3-sqrt(5))^n + (3+sqrt(5))^n.

a(n) = 6*a(n-1) - 4*a(n-2) for n>1.

(End)