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

1,5

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.

Links

Table of n, a(n) for n=1..39.

Mathematica

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

Prog

(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))

Crossrefs

Cf. A349904, A000073.

Sequence in context: A078172 A022308 A278137 * A206268 A178749 A121980

Adjacent sequences: A349974 A349975 A349976 * A349978 A349979 A349980

Keyword

nonn

Author

Peter Luschny, Dec 07 2021

Status

approved