A396582 - OEIS

%I #19 May 30 2026 10:31:54

%S 1,6,-8,40,-112,384,-1216,3968,-12800,41472,-134144,434176,-1404928,

%T 4546560,-14712832,47611904,-154075136,498597888,-1613496320,

%U 5221384192,-16896753664,54679044096,-176945102848,572606382080,-1852993175552,5996411879424,-19404796461056

%N a(n) = 4*a(n - 2) - 2*a(n - 1) for n >= 2, a(0) = 1, a(1) = 6.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,4).

%F a(n) = -(-2)^n * (F(n) + L(n-1)) where F ~ Fibonacci A000045 and L ~ Lucas A000032.

%F a(n) = [x^n] (1 + 8*x) / (1 + 2*x - 4*x^2).

%F a(n) = n! * [x^n] exp(-x)*(7*sinh(r*x)/r + cosh(r*x)) where r = sqrt(5).

%F a(n) = ((1 - 7/r) * (-r - 1)^n + (1 + 7/r) * (r - 1)^n) / 2 where r = sqrt(5).

%F 2^n * a(-n) = A000285(n).

%F a(n) / 2^n = A396201(n).

%p a := n -> ((1 - 7/sqrt(5)) * (-sqrt(5) - 1)^n + (1 + 7/sqrt(5)) * (sqrt(5) - 1)^n )/2: seq(simplify(a(n)), n = 0..26);

%t LinearRecurrence[{-2,4},{1,6},27] (* _James C. McMahon_, May 30 2026 *)

%o (Python)

%o def A396582gen(bound):

%o     a, b = 1, 6

%o     if bound >= 1: yield a

%o     if bound >= 2: yield b

%o     for _ in range(2, bound):

%o         c = 4 * a - 2 * b

%o         yield c

%o         a, b = b, c

%o for v in A396582gen(27): print(v, end=", ")

%o (SageMath)

%o a = BinaryRecurrenceSequence(-2, 4, 1, 6)

%o print([a(i) for i in range(27)])

%Y Cf. A396201, A000032 (Lucas), A000045 (Fibonacci), A000285.

%K sign,easy

%O 0,2

%A _Peter Luschny_, May 29 2026