A347178 -id:A347178 - cp's OEIS frontend

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211

Offset: 0

Vaclav Kotesovec, Sep 29 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

This is the sum of the imaginary parts of the complex roots of the cubic equation 8*r^3 + 2*r - 1 = 0 , and its real solution is A347178. - Gerry Martens, Apr 02 2024

			1.1615413999972519360879176872471740748431...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347178.

RealDigits[(1/12) 2^(2/3) 3^(5/6) ((Sqrt[93] - 9)^(1/3) + (9 + Sqrt[93])^(1/3)), 10, 110][[1]]

(1/12)*2^(2/3)*3^(5/6)*((sqrt(93) - 9)^(1/3) + (9 + sqrt(93))^(1/3)) \\ Michel Marcus, Aug 21 2021

2*imag(polroots(8*x^3 + 2*x - 1)[3]) \\ Gerry Martens, Apr 02 2024

Equals cosh(asinh(3*sqrt(3)/2)/3). - Gerry Martens, Apr 02 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

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Python

Python

Formula