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
Examples
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;
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,8).
Programs
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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
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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] -
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)]
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.
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