A295850 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = 0, a(2) = 2, a(3) = 1.

0, 0, 2, 1, 7, 6, 21, 23, 60, 75, 167, 226, 457, 651, 1236, 1823, 3315, 5010, 8837, 13591, 23452, 36531, 62031, 97538, 163665, 259155, 431012, 686071, 1133467, 1811346, 2977581, 4772543, 7815660, 12555435, 20502167, 32992066, 53756377, 86617371, 140898036

Offset

0,3

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Therefore the terms are all >= 0. - Georg Fischer, Feb 19 2019

Links

Clark Kimberling, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (1, 3, -2, -2)

Formula

a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 2*a(n-4), where a(0) = 0, a(1) = 0, a(2) = 2, a(3) = 1. [Corrected by Colin Barker, Dec 15 2017]

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

From Colin Barker, Dec 15 2017: (Start)

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

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

(End)

Mathematica

LinearRecurrence[{1, 3, -2, -2}, {0, 0, 2, 1}, 100]

Crossrefs

Cf. A001622, A000045.

Sequence in context: A257577 A060583 A246751 * A078104 A072280 A217106

Adjacent sequences: A295847 A295848 A295849 * A295851 A295852 A295853

Keyword

easy,nonn

Author

Clark Kimberling, Dec 01 2017

Status

approved