A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011

Offset

0,1

Comments

Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (4,1).

Formula

a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.

G.f.: (3-x)/(1-4*x-x^2).

a(n) = A033887(n) + A014445(n+1).

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

a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - Wolfdieter Lang, Mar 09 2011.

a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Dec 10 2015

Maple

with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011

Mathematica

Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* Alonso del Arte, Nov 29 2010 *)
LinearRecurrence[{4, 1}, {3, 11}, 30] (* Harvey P. Dale, Apr 14 2021 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009
(Magma) [Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
(PARI) Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

Crossrefs

Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).

Sequence in context: A290712 A248208 A112567 * A151142 A151143 A151144

Adjacent sequences: A163060 A163061 A163062 * A163064 A163065 A163066

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

Status

approved