A347177 -id:A347177 - cp's OEIS frontend

A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

3, 4, 1, 1, 6, 3, 9, 0, 1, 9, 1, 4, 0, 0, 9, 6, 6, 3, 6, 8, 4, 7, 4, 1, 8, 6, 9, 8, 5, 5, 5, 2, 4, 1, 2, 8, 4, 4, 5, 5, 9, 4, 2, 9, 0, 9, 4, 8, 9, 9, 9, 2, 8, 8, 9, 0, 1, 8, 6, 4, 3, 0, 3, 3, 1, 9, 9, 4, 8, 3, 3, 9, 3, 4, 3, 4, 9, 9, 0, 1, 0, 5, 4, 0, 8, 6, 6, 0, 2, 1, 8, 9, 3, 1, 0, 2, 5, 6, 4, 1, 4, 7, 7, 9, 6, 6, 5, 9, 3, 8

Offset: 0

Ilya Gutkovskiy, Aug 21 2021

			0.3411639019140096636847418698555241284...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347177.

RealDigits[((29 + 3 Sqrt[93])^(1/3) - 2^(1/3))/(2 (3 (9 + Sqrt[93]))^(1/3)), 10, 110][[1]]

((29 + 3*sqrt(93))^(1/3) - 2^(1/3))/(2*(3*(9 + sqrt(93)))^(1/3)) \\ Michel Marcus, Aug 21 2021

Equals -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3)/sqrt(3). - Vaclav Kotesovec, Sep 29 2024

A355141 a(0)=1, a(1)=a(2)=0; for n > 2, a(n) = a(n-1) + s if n is odd, a(n-1) - s if n is even, where s = a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, -1, -1, 0, -2, 1, 0, 1, 3, -1, 2, -2, -3, 0, -5, 3, 1, 2, 8, -3, 4, -5, -9, 1, -12, 8, 5, 4, 21, -9, 7, -12, -26, 5, -28, 21, 19, 7, 54, -26, 9, -28, -73, 19, -63, 54, 64, 9, 136, -73, -1, -63, -200, 64, -135, 136, 201, -1, 335, -200, -66

Offset: 0

Nate Shaw, Jun 20 2022

Apparently, |a(n)|^(1/n) < 1.100276355... = (A347177^2 + A347178^2)^(1/4). - Jon E. Schoenfield, Jun 22 2022

			For n = 12, a(12) = 1: The previous three terms are a(9) = -2, a(10) = 1, a(11) = 0, whose sum is -1. 12 is even, so we subtract that sum from the previous term, a(11) = 0, which gives a(12) = 0 - (-1) = 1.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1,0,-1).

CoefficientList[Series[(1 + x^3 + x^4 + x^5)/(1 + x^4 + x^6), {x, 0, 65}], x] (* Michael De Vlieger, Jun 22 2022 *)
LinearRecurrence[{0,0,0,-1,0,-1},{1,0,0,1,0,1},70] (* Harvey P. Dale, Aug 11 2025 *)

def a(n):
    if n in [0, 1, 2]:
        return [1, 0, 0][n]
    else:
        previous_terms = [1, 0, 0]
        for i in range(n - 2):
            previous_three_terms = previous_terms[-3:]
            previous_three_terms_sum = sum(previous_three_terms)
            current_term = previous_terms[-1]
            if i % 2 == 0:
                previous_terms.append(current_term + previous_three_terms_sum)
            else:
                previous_terms.append(current_term - previous_three_terms_sum)
        return previous_terms[-1]
print([a(n) for n in range(66)])

a, terms = [1, 0,  0], 66
[a.append(a[-1]-(-1)**(n%2)*sum(a[-3:])) for n in range(3, terms)]
print(a) # Michael S. Branicky, Jun 22 2022

a(n) = a(n-1) + (2*(n mod 2) - 1)*(a(n-3) + a(n-2) + a(n-1)), with a(0) = 1, a(1) = 0, and a(2) = 0.

G.f.: (1 + x^3 + x^4 + x^5)/(1 + x^4 + x^6). - Stefano Spezia, Jun 22 2022

cp's OEIS Frontend

A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

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