A323232 a(n) = 2^n*J(n, 1/2) where J(n, x) are the Jacobsthal polynomials as defined in A322942.

1, 3, 9, 51, 225, 1083, 5049, 23811, 111825, 525963, 2472489, 11625171, 54655425, 256967643, 1208146329, 5680180131, 26705711025, 125558574123, 590321410569, 2775432824691, 13048869758625, 61350071873403, 288441173689209, 1356124096054851, 6375901677678225

Offset

0,2

Comments

Is it true that p prime and p not 2 or 5 implies that a(p) is squarefree?

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,8).

Formula

a(n) = 3*a(n-1) + 8*a(n-2) for n >= 3.

a(n) is an odd integer and 3 | a(n) if n > 0.

a(n) = Sum_{k=0..n} 2^(n - k)*A322942(n, k).

a(n) = [x^n] (8*x^2 - 1)/(8*x^2 + 3*x - 1).

Let s = sqrt(41), u = -1/(s+3) and v = 1/(s-3); then

a(n) = (3/s)*16^n*(v^n - u^n) for n >= 1.

a(n) = n! [x^n](1 + (6*exp(3*x/2)*sinh(s*x/2))/s).

a(n) = n! [x^n](1 + (3/s)*(exp((3 + s)*x/2) - exp((3 - s)*x/2))).

a(n)/a(n+1) -> 2/(sqrt(41) + 3) = (sqrt(41) - 3)/16 for n -> oo.

Example

The first few prime factorizations of a(n):
   1| 3;
   2| 3^2;
   3| 3   * 17;
   4| 3^2 * 5^2;
   5| 3   * 19^2;
   6| 3^3 * 11 * 17;
   7| 3   * 7937;
   8| 3^2 * 5^2 * 7 * 71;
   9| 3   * 17 * 10313;
  10| 3^2 * 19^2 * 761;
  11| 3   * 3875057;
  12| 3^3 * 5^2 * 11 * 17 * 433;
  13| 3   * 85655881;
  14| 3^2 * 13 * 1301 * 7937;
  15| 3   * 17 * 19^2 * 308521;
  16| 3^2 * 5^2 * 7 * 71 * 79 * 3023;
  17| 3   * 67 * 624669523;
  18| 3^4 * 11 * 17 * 3779 * 10313;
  19| 3   * 419 * 2207981563;

Maple

a := proc(n) option remember:
    if n < 3 then return [1, 3, 9][n+1] fi;
    8*a(n-2) + 3*a(n-1) end:
seq(a(n), n=0..24);

Mathematica

LinearRecurrence[{3, 8}, {1, 3, 9}, 25]

Prog

(Sage)
def a():
    yield 1
    yield 3
    c = 3; b = 9
    while True:
        yield b
        a = (b << 2) + (c << 3) - b
        c = b
        b = a
A323232 = a()
[next(A323232) for _ in range(30)]
(Magma) [1] cat [n le 2 select 3^n else 3*Self(n-1) +8*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 27 2021

Crossrefs

Cf. A015525, A322942.

Sequence in context: A261542 A079836 A171951 * A366932 A319105 A106329

Adjacent sequences: A323229 A323230 A323231 * A323233 A323234 A323235

Keyword

nonn,easy

Author

Peter Luschny, Jan 07 2019

Status

approved