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

0,1

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).

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)

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 *)

Prog

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

Crossrefs

Cf. A014448, A108404, A098648.

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

Sequence in context: A302336 A225877 A350524 * A218941 A303829 A272820

Adjacent sequences: A228839 A228840 A228841 * A228843 A228844 A228845

Keyword

nonn,easy

Author

R. J. Mathar, Nov 10 2013

Status

approved