A047081 a(n) = Sum_{k=0..n} T(n, k), array T as in A047080.

1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423, 778, 1431, 2632, 4841, 8904, 16377, 30122, 55403, 101902, 187427, 344732, 634061, 1166220, 2145013, 3945294, 7256527, 13346834, 24548655, 45152016, 83047505, 152748176, 280947697, 516743378, 950439251, 1748130326, 3215312955, 5913882532, 10877325813

Offset: 0

Author

Clark Kimberling

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A047080, A047082, A047083, A047084, A047085, A047086, A047087, A047088.

Cf. A000073, A254006.

Programs

Magma

[n le 3 select n else Self(n-1) +Self(n-2) +Self(n-3): n in [1..60]]; // G. C. Greubel, Oct 31 2022

Mathematica

LinearRecurrence[{1,1,1}, {1,2,3}, 61] (* G. C. Greubel, Oct 31 2022 *)

SageMath

@CachedFunction
def a(n): return (n+1) if (n<3) else a(n-1) +a(n-2) +a(n-3) # a = A047081
[a(n) for n in (0..60)] # G. C. Greubel, Oct 31 2022

Formula

From G. C. Greubel, Oct 31 2022: (Start)

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

a(n) = A000073(n+1) + A000073(n+2). (End)

a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-3*r^2+5*r+2). - Fabian Pereyra, Nov 23 2024

a(n) = A001590(n+3). - R. J. Mathar, Mar 28 2025

Extensions

Data corrected by G. C. Greubel, Oct 31 2022