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A082762 Trinomial transform of Lucas numbers (A000032). 6
1, 8, 44, 232, 1216, 6368, 33344, 174592, 914176, 4786688, 25063424, 131233792, 687149056, 3597959168, 18839158784, 98643116032, 516502061056, 2704439902208, 14160631169024, 74146027405312, 388233639755776, 2032817728913408, 10643971814457344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-4).

FORMULA

a(n) = Sum_{k=0..2*n} Trinomial(n,k)*Lucas(k+1), where Trinomial(n,k) = trinomial coefficients (A027907).

a(n) = 2^n*Lucas(2*n+1), where Lucas = A000032.

From Philippe Deléham, Mar 01 2004: (Start)

a(n) = 2^n*A002878(n) = 2^(-n)*Sum_{k>=0} C(2*n+1,2*k)*5^k; see A091042.

a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1). (End)

From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009: (Start)

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

Third binomial transform of 1, 5, 5, 25, 25, 125. (End)

G.f.: (1 + 2*x)/(1 - 6*x + 4*x^2). - Colin Barker, Mar 23 2012

MATHEMATICA

a[n_]:=(MatrixPower[{{2, 2}, {2, 4}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* Robert G. Wilson v, Mar 07 2011 *)

LinearRecurrence[{6, -4}, {1, 8}, 30] (* G. C. Greubel, Dec 21 2017 *)

PROG

(PARI) x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ G. C. Greubel, Dec 21 2017

(MAGMA) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)-4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CROSSREFS

Cf. A000032, A027907, A091042, A292277.

Sequence in context: A270935 A197213 A198768 * A249794 A322498 A147828

Adjacent sequences:  A082759 A082760 A082761 * A082763 A082764 A082765

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, May 21 2003

STATUS

approved

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Last modified September 27 12:01 EDT 2020. Contains 337380 sequences. (Running on oeis4.)