A100233 a(n) = Lucas(3*n) - 1.

1, 3, 17, 75, 321, 1363, 5777, 24475, 103681, 439203, 1860497, 7881195, 33385281, 141422323, 599074577, 2537720635, 10749957121, 45537549123, 192900153617, 817138163595, 3461452808001, 14662949395603, 62113250390417, 263115950957275, 1114577054219521

Offset

0,2

Comments

Main diagonal of triangle A100232.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Formula

a(n) = A014448(n) - 1.

a(n) = 4*a(n-1) + a(n-2) + 4 for n>1, with a(0)=1, a(1)=3.

G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-4*x-x^2)).

a(n) = [x^n] ( 1 + 2*x + sqrt(1 + 2*x + 5*x^2) )^n. Cf. A016064. - Peter Bala, Jun 23 2015

From Colin Barker, Jun 02 2016: (Start)

a(n) = -1+(2-sqrt(5))^n+(2+sqrt(5))^n.

a(n) = 5*a(n-1)-3*a(n-2)-a(n-3) for n>2.

G.f.: (1-2*x+5*x^2) / ((1-x)*(1-4*x-x^2)).

(End)

Mathematica

Table[LucasL[3*n] - 1, {n, 0, 50}] (* or *) LinearRecurrence[{5, -3, -1}, {1, 3, 17}, 30] (* G. C. Greubel, Dec 21 2017 *)

Prog

(PARI) a(n)=if(n==0, 1, n*polcoeff(log((1-x)/(1-4*x-x^2)+x*O(x^n)), n))
(PARI) Vec((1-2*x+5*x^2)/((1-x)*(1-4*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 02 2016
(Magma) I:=[1, 3, 17]; [n le 3 select I[n] else 5*Self(n-1) -3*Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Crossrefs

Cf. A000032, A100230, A100232, A016064.

Sequence in context: A142868 A171875 A356390 * A015525 A062224 A217958

Adjacent sequences: A100230 A100231 A100232 * A100234 A100235 A100236

Keyword

nonn,easy

Author

Paul D. Hanna, Nov 29 2004

Extensions

New definition from Ralf Stephan, Dec 01 2004

Status

approved