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A155117 a(n) = 4*a(n-1) + 4*a(n-2), n>2, a(0)=1, a(1)=3, a(2)=15. 10
1, 3, 15, 72, 348, 1680, 8112, 39168, 189120, 913152, 4409088, 21288960, 102792192, 496324608, 2396467200, 11571167232, 55870537728, 269766819840, 1302549430272, 6289265000448, 30367257722880, 146626090893312 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

G.f.: (1-x-x^2)/(1-4*x-4*x^2) . a(n)=3*A086347(n), n>=1 .

a(n) = (3/2)*( (2+2*sqrt(2))^(n-1) + (2-2*sqrt(2))^(n-1) ) + (9/8)*sqrt(2)*( (2+2*sqrt(2))^(n-1) - (2-2*sqrt(2))^(n-1) ) + (1/4)*[binomial(2*n,n) mod 2], with n>=0. - Paolo P. Lava, Jan 26 2009

From G. C. Greubel, Mar 25 2021: (Start)

a(n) = (1/4)*[n=0] - 3*(2*i)^(n-2)*ChebyshevU(n, -i).

a(n) = (1/4)*[n=0] + 3*2^(n-2)*P_{n+1}, where P_{n} = A000129(n) (Pell numbers). (End)

MAPLE

1, seq(simplify(-3*(2*I)^(n-2)*ChebyshevU(n, -I)), n = 1..30); # G. C. Greubel, Mar 25 2021

MATHEMATICA

With[{m=4}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

PROG

(Magma) m:=4; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

(Sage) m=4; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

CROSSREFS

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), this sequence (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

Cf. A000129.

Sequence in context: A183547 A123942 A290902 * A137638 A156019 A145839

Adjacent sequences:  A155114 A155115 A155116 * A155118 A155119 A155120

KEYWORD

nonn

AUTHOR

Philippe Deléham, Jan 20 2009

STATUS

approved

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Last modified April 12 12:39 EDT 2021. Contains 342920 sequences. (Running on oeis4.)