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%I #15 Sep 08 2022 08:46:23
%S 1,3,9,51,225,1083,5049,23811,111825,525963,2472489,11625171,54655425,
%T 256967643,1208146329,5680180131,26705711025,125558574123,
%U 590321410569,2775432824691,13048869758625,61350071873403,288441173689209,1356124096054851,6375901677678225
%N a(n) = 2^n*J(n, 1/2) where J(n, x) are the Jacobsthal polynomials as defined in A322942.
%C Is it true that p prime and p not 2 or 5 implies that a(p) is squarefree?
%H G. C. Greubel, <a href="/A323232/b323232.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,8).
%F a(n) = 3*a(n-1) + 8*a(n-2) for n >= 3.
%F a(n) is an odd integer and 3 | a(n) if n > 0.
%F a(n) = Sum_{k=0..n} 2^(n - k)*A322942(n, k).
%F a(n) = [x^n] (8*x^2 - 1)/(8*x^2 + 3*x - 1).
%F Let s = sqrt(41), u = -1/(s+3) and v = 1/(s-3); then
%F a(n) = (3/s)*16^n*(v^n - u^n) for n >= 1.
%F a(n) = n! [x^n](1 + (6*exp(3*x/2)*sinh(s*x/2))/s).
%F a(n) = n! [x^n](1 + (3/s)*(exp((3 + s)*x/2) - exp((3 - s)*x/2))).
%F a(n)/a(n+1) -> 2/(sqrt(41) + 3) = (sqrt(41) - 3)/16 for n -> oo.
%e The first few prime factorizations of a(n):
%e 1| 3;
%e 2| 3^2;
%e 3| 3 * 17;
%e 4| 3^2 * 5^2;
%e 5| 3 * 19^2;
%e 6| 3^3 * 11 * 17;
%e 7| 3 * 7937;
%e 8| 3^2 * 5^2 * 7 * 71;
%e 9| 3 * 17 * 10313;
%e 10| 3^2 * 19^2 * 761;
%e 11| 3 * 3875057;
%e 12| 3^3 * 5^2 * 11 * 17 * 433;
%e 13| 3 * 85655881;
%e 14| 3^2 * 13 * 1301 * 7937;
%e 15| 3 * 17 * 19^2 * 308521;
%e 16| 3^2 * 5^2 * 7 * 71 * 79 * 3023;
%e 17| 3 * 67 * 624669523;
%e 18| 3^4 * 11 * 17 * 3779 * 10313;
%e 19| 3 * 419 * 2207981563;
%p a := proc(n) option remember:
%p if n < 3 then return [1, 3, 9][n+1] fi;
%p 8*a(n-2) + 3*a(n-1) end:
%p seq(a(n), n=0..24);
%t LinearRecurrence[{3, 8}, {1, 3, 9}, 25]
%o (Sage)
%o def a():
%o yield 1
%o yield 3
%o c = 3; b = 9
%o while True:
%o yield b
%o a = (b << 2) + (c << 3) - b
%o c = b
%o b = a
%o A323232 = a()
%o [next(A323232) for _ in range(30)]
%o (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
%Y Cf. A015525, A322942.
%K nonn,easy
%O 0,2
%A _Peter Luschny_, Jan 07 2019
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