A295860 a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4), where a(0) = -2, a(1) = 1, a(2) = 0, a(3) = 1.

-2, 1, 0, 1, 3, 4, 11, 15, 34, 49, 99, 148, 279, 427, 770, 1197, 2095, 3292, 5643, 8935, 15090, 24025, 40139, 64164, 106351, 170515, 280962, 451477, 740631, 1192108, 1949123, 3141231, 5123122, 8264353, 13453011, 21717364, 35301447, 57018811, 92582402

Offset: 0

Author

Clark Kimberling, Jan 07 2018

Keywords

• easy
• sign

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).

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)

Crossrefs

Cf. A001622, A000045, A016116.

Programs

Maple

f:= n -> - 2^floor((n+1)/2) + 4*combinat:-fibonacci(n) - combinat:-fibonacci(n+1):
map(f, [$0..30]); # Robert Israel, Jan 12 2018

Mathematica

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

Formula

G.f.: (-2 + 3 x + 5 x^2 - 6 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

a(n) = -A016116(n+1) + 4*A000045(n) - A000045(n+1). - Robert Israel, Jan 12 2018