A295690 a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 2, a(2) = 1, a(3) = 1.

2, 2, 1, 1, 5, 8, 10, 16, 29, 47, 73, 118, 194, 314, 505, 817, 1325, 2144, 3466, 5608, 9077, 14687, 23761, 38446, 62210, 100658, 162865, 263521, 426389, 689912, 1116298, 1806208, 2922509, 4728719, 7651225, 12379942, 20031170, 32411114, 52442281, 84853393

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, A000032.

Programs

Magma

a:=[2,2,1,1]; [n le 4 select a[n] else Self(n-1) + Self(n-3) + Self(n-4):n in [1..40]]; // Marius A. Burtea, Nov 13 2019

Mathematica

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

Formula

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

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

From Peter Bala, Nov 12 2019: (Start)

a(2*n) = (3/5)*Lucas(2*n) + (4/5)*(-1)^n.

a(2*n+1) = (3/5)*Lucas(2*n+1) + (7/5)*(-1)^n.

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

a(2*n+1) = a(2*n) + a(2*n-1) + 2*(-1)^n.

a(2*n+1)*F(n+3) - a(2*n+3)*F(n-1) = 3*F(n+1)^3, where F(n) = A000045(n). (End)