A349977 Inverse Euler transform of the classical tribonacci numbers.

0, 1, 1, 1, 3, 4, 8, 13, 23, 38, 68, 114, 201, 343, 600, 1037, 1817, 3157, 5543, 9692, 17047, 29952, 52828, 93157, 164743, 291459, 516679, 916626, 1628684, 2896261, 5156925, 9189769, 16393652, 29268223, 52300907, 93529331, 167390342, 299787639, 537281476

Offset: 1

Author

Peter Luschny, Dec 07 2021

Keywords

• nonn

Comments

The classical tribonacci numbers are defined a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = 0 and a(1) = a(2) = 1.

See A349904 for the analogous sequence for the shifted tribonacci numbers A000073.

Crossrefs

Cf. A349904, A000073.

Programs

Mathematica

(* EulerInvTransform is defined in A022562. *)
EulerInvTransform[LinearRecurrence[{1, 1, 1}, {0, 1, 1}, 40]]

Python

# After the Maple program of Alois P. Heinz in A349904.
from functools import cache
from math import comb
def euler_inv_trans(a: callable, len: int):
    @cache
    def h(n: int, k: int):
        if n == 0: return 1
        if k <  1: return 0
        bk = b(k)
        R = range(int(bk == 0), 1 + n // k)
        return sum(comb(bk + j - 1, j) * h(n - k * j, k - 1) for j in R)
    @cache
    def b(n: int): return a(n - 1) - h(n, n - 1)
    return [b(n) for n in range(1, len)]
@cache
def tribonacci(n: int):
    return sum(tribonacci(n - j - 1) for j in range(3)) if n >= 3 else min(n, 1)
print(euler_inv_trans(tribonacci, 40))