A085480 Expansion of 3*x*(1+2*x)/(1-3*x-3*x^2).

0, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079

Offset

1,2

Comments

A Jacobsthal variation.

p - q = sqrt(21); p*q = -3; p + q = 3.

References

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (3,3).

Formula

a(n) = p^n + q^n, where p = (3 + sqrt(21))/2, q = (3 - sqrt 21)/2.

a(n) = 3*a(n-1) + 3*a(n-2), a(1)=3, a(2)=15. - Philippe Deléham, Nov 19 2008

G.f.: G(0)/x - 2/x, where G(k) = 1 + 1/(1 - x*(7*k-3)/(x*(7*k+4) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013

Example

a(4) = q^4 + q^4 = 207; p^5 + q^5 = 783, where p = (3 + sqrt(21))/2, q = (3 - sqrt(21))/2.

Mathematica

CoefficientList[Series[3x (1+2x)/(1-3x-3x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 3}, {0, 3, 15}, 30] (* Harvey P. Dale, Jan 10 2021 *)

Crossrefs

Cf. A030195.

Sequence in context: A290764 A286986 A261565 * A265974 A099581 A026696

Adjacent sequences: A085477 A085478 A085479 * A085481 A085482 A085483

Keyword

nonn,easy

Author

Gary W. Adamson, Jul 02 2003

Extensions

Zero prepended by Harvey P. Dale, Jan 10 2021

Status

approved