The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A120718 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A120718

Expansion of 3*x/(1 - 2*x^2 - 2*x + x^3).
(history; published version)

#34 by Harvey P. Dale at Fri Sep 06 14:49:03 EDT 2024

STATUS

editing

approved

#33 by Harvey P. Dale at Fri Sep 06 14:49:01 EDT 2024

MATHEMATICA

CoefficientList[Series[3x/(1-2x^2-2x+x^3), {x, 0, 30}], x] (* Harvey P. Dale, Sep 06 2024 *)

STATUS

approved

editing

#32 by Alois P. Heinz at Fri Jul 21 04:22:22 EDT 2023

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reviewed

approved

#31 by Michel Marcus at Fri Jul 21 03:31:51 EDT 2023

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proposed

reviewed

#30 by G. C. Greubel at Fri Jul 21 02:57:49 EDT 2023

STATUS

editing

proposed

#29 by G. C. Greubel at Fri Jul 21 02:56:42 EDT 2023

FORMULA

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

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

(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 *)

PROG

(PARI) a(n) = round((-3)*(fibonacci(2^(-1-n)*((-1)^n*+2^(1+n) +(3-sqrt fibonacci(5))^2*n*(-1+sqrt(5)) - (-1+sqrt(5))*(3+sqrt(5))^n))/5) \\ 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.

STATUS

approved

editing

Discussion

Fri Jul 21

02:57

G. C. Greubel: Placed Pari into a form that removed the need of "round".

#28 by Joerg Arndt at Sat Oct 01 06:24:42 EDT 2016

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reviewed

approved

#27 by Michel Marcus at Sat Oct 01 06:15:40 EDT 2016

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proposed

reviewed

#26 by Colin Barker at Sat Oct 01 06:11:43 EDT 2016

STATUS

editing

proposed

#25 by Colin Barker at Sat Oct 01 06:11:24 EDT 2016

LINKS

Colin Barker, <a href="/A120718/b120718.txt">Table of n, a(n) for n = 0..1000</a>

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, -1).

FORMULA

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

(End)

PROG

(PARI) a(n) = round((-3)*(2^(-1-n)*((-1)^n*2^(1+n)+(3-sqrt(5))^n*(-1+sqrt(5))-(1+sqrt(5))*(3+sqrt(5))^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

STATUS

approved

editing