A120718 Expansion of 3*x/(1 - 2*x^2 - 2*x + x^3).

0, 3, 6, 18, 45, 120, 312, 819, 2142, 5610, 14685, 38448, 100656, 263523, 689910, 1806210, 4728717, 12379944, 32411112, 84853395, 222149070, 581593818, 1522632381, 3986303328, 10436277600, 27322529475, 71531310822, 187271402994, 490282898157, 1283577291480

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

a(n) = 3*A001654(n). - Arkadiusz Wesolowski, Sep 15 2012

From Colin Barker, Oct 01 2016: (Start)

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n>2.

a(n) = (3/2^(n+1))*( (1-sqrt(5))*(3-sqrt(5))^n + (1+sqrt(5))*(3+sqrt(5))^n + (-2)^(n+1) )/5. (End)

a(n) = (3/5)*(Lucas(2*n+1) - (-1)^n). - G. C. Greubel, Jul 21 2023

Mathematica

LinearRecurrence[{2, 2, -1}, {0, 3, 6}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
CoefficientList[Series[3x/(1-2x^2-2x+x^3), {x, 0, 30}], x] (* Harvey P. Dale, Sep 06 2024 *)

Prog

(PARI) a(n) = 3*(fibonacci(2*n+2) + fibonacci(2*n) - (-1)^n)/5 \\ Colin Barker, Oct 01 2016
(PARI) concat(0, Vec(3*x/(1-2*x^2-2*x+x^3) + O(x^40))) \\ Colin Barker, Oct 01 2016
(Magma) [(3/5)*(Lucas(2*n+1) -(-1)^n): n in [0..40]]; // G. C. Greubel, Jul 21 2023
(SageMath) [(3/5)*(lucas_number2(2*n+1, 1, -1) -(-1)^n) for n in range(41)] # G. C. Greubel, Jul 21 2023

Crossrefs

Cf. A000032, A000045, A001654, A001906, A072845.

Sequence in context: A354387 A121188 A331678 * A032120 A115344 A223044

Adjacent sequences: A120715 A120716 A120717 * A120719 A120720 A120721

Keyword

nonn,easy

Author

Roger L. Bagula, Aug 13 2006

Extensions

Offset corrected by Arkadiusz Wesolowski, Sep 15 2012

Status

approved