A346461 a(n) = 2^A042965(n+1).

2, 8, 16, 32, 128, 256, 512, 2048, 4096, 8192, 32768, 65536, 131072, 524288, 1048576, 2097152, 8388608, 16777216, 33554432, 134217728, 268435456, 536870912, 2147483648, 4294967296, 8589934592, 34359738368, 68719476736, 137438953472, 549755813888, 1099511627776

Offset

1,1

Comments

For each n, all positive integers cannot be colored with two colors without any positive integer x being the same color as 4*x or a(n)*x.

A346459(a(n), 4) = 1 for all n > 1.

{a(n)} is the fourth row in A346460.

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (0,0,16).

Formula

a(n) = 2^(floor((4*n+1)/3)).

a(n) = a(n-1) * a(n-3) / a(n-4) for n > 4.

From Stefano Spezia, Sep 04 2021: (Start)

G.f.: 2*x*(1 + 4*x + 8*x^2)/(1 - 16*x^3).

a(n) = 16*a(n-3) for n > 3. (End)

Example

All positive integers cannot be colored with two colors without any positive integer x being the same color as a(1)*x = 2*x or 4*x, since at least two elements of the set {2, 4, 8} have to be the same color, which violates the condition.

Mathematica

Rest@ CoefficientList[Series[2 x (1 + 4 x + 8 x^2)/(1 - 16 x^3), {x, 0, 30}], x] (* Michael De Vlieger, Oct 12 2021 *)

Prog

(PARI) A346461(n) = (1<<((4*(1+n)-3)\3)); \\ Antti Karttunen, Nov 11 2021
(Python)
print([2 ** ((4 * n + 1) // 3) for n in range(1, 31)])

Crossrefs

Cf. A000302, A004171, A016825, A042965, A346459, A346460.

Sequence in context: A077666 A232358 A212318 * A232392 A176143 A296946

Adjacent sequences: A346458 A346459 A346460 * A346462 A346463 A346464

Keyword

nonn,easy

Author

M. Eren Kesim, Sep 04 2021

Status

approved