A383873 a(n) = 3*a(n-1) - 2*a(n-2) + 5*a(n-3) starting with 1, 2, 3.

1, 2, 3, 10, 34, 97, 273, 795, 2324, 6747, 19568, 56830, 165089, 479447, 1392313, 4043490, 11743079, 34103822, 99042758, 287636025, 835341669, 2425966747, 7045397028, 20460965935, 59421937484, 172570865722, 501173551873, 1455488611595, 4226973059649

Offset: 0

Author

Raul Prisacariu, May 18 2025

Keywords

• nonn
• easy

Comments

The sequence appears as an example of recurring series in "A Mathematical and Philosophical Dictionary" by Charles Hutton.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Charles Hutton, Recurring Series, A mathematical and philosophical dictionary Vol. 2 (1795), 440.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,5).

Programs

Magma

I:=[ 1, 2, 3]; [n le 3 select I[n] else 3*Self(n-1)-2*Self(n-2)+5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 21 2025

Mathematica

LinearRecurrence[{3,-2,5},{1,2,3},30] (* Vincenzo Librandi, May 21 2025 *)

Formula

G.f.: (x^2+x-1)/(5*x^3-2*x^2+3*x-1). - Alois P. Heinz, May 19 2025