A295683 a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 1, a(2) = 0, a(3) = 1.

2, 1, 0, 1, 4, 5, 6, 11, 20, 31, 48, 79, 130, 209, 336, 545, 884, 1429, 2310, 3739, 6052, 9791, 15840, 25631, 41474, 67105, 108576, 175681, 284260, 459941, 744198, 1204139, 1948340, 3152479, 5100816, 8253295, 13354114, 21607409, 34961520, 56568929, 91530452

Offset: 0

Author

Clark Kimberling, Nov 29 2017

Keywords

• nonn
• easy

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 (1,0,1,1).

Crossrefs

Cf. A001622, A000045, A056594.

Programs

Magma

I:=[2,1,0,1]; [n le 4 select I[n] else Self(n-1) +Self(n-3) +Self(n-4): n in [1..45]]; // G. C. Greubel, Apr 03 2019

Maple

f:= gfun:-rectoproc(a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = 2, a(1) = 1, a(2) = 0, a(3) = 1}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Apr 03 2019

Mathematica

LinearRecurrence[{1, 0, 1, 1}, {2, 1, 0, 1}, 45]
CoefficientList[Series[(2-x-x^2-x^3)/(1-x-x^3-x^4),{x,0,40}],x] (* Harvey P. Dale, Jun 22 2025 *)

PARI

my(x='x+O('x^45)); Vec((-2 + x + x^2 + x^3)/(-1 + x + x^3 + x^4)) \\ Georg Fischer, Apr 03 2019

Sage

((2-x-x^2-x^3)/(1-x-x^3-x^4)).series(x, 45).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

Formula

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

a(n) = (2/5) * A000045(n) + (4/5) * A000045(n-1) + (6/5) * A056594(n) + (3/5) * A056594(n-1) for n >= 1. - Robert Israel, Apr 03 2019

Extensions

a(0) = 2 amended by Georg Fischer, Apr 03 2019