A347861 - OEIS

%I #24 Jan 25 2022 21:01:43

%S 2,3,5,6,5,24,5,71,5,194,5,516,5,1359,5,3566,5,9344,5,24471,5,64074,5,

%T 167756,5,439199,5,1149846,5,3010344,5,7881191,5,20633234,5,54018516,

%U 5,141422319,5,370248446,5,969323024,5,2537720631,5,6643838874,5,17393795996,5,45537549119,5,119218851366,5

%N a(n) = A000032(n)*A000032(n+1) mod A000032(n+2).

%C The analogous sequence for Fibonacci numbers instead of Lucas numbers is A333599.

%H Robert Israel, <a href="/A347861/b347861.txt">Table of n, a(n) for n = 0..4760</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3,-1,-1).

%F G.f.: 4*x - 3 - (x + 3)/(2*(x^2 + x - 1)) - (x - 3)/(2*(x^2 - x - 1)) + 5/(x + 1).

%F a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n >= 7.

%F a(n) = 5 for even n >= 2.

%F a(n) = A000032(n+2)-5 for odd n >= 3.

%e a(3) = A000032(3)*A000032(4) mod A000032(5) = 4*7 mod 11 = 6.

%p L:= n -> combinat:-fibonacci(n-1)+combinat:-fibonacci(n+1):

%p f:= n -> L(n)*L(n+1) mod L(n+2):

%p map(f, [$0..40]);

%t With[{L = LucasL}, Table[Mod[L[n]*L[n + 1], L[n + 2]], {n, 0, 50}]] (* _Amiram Eldar_, Jan 24 2022 *)

%o (PARI) L(n) = fibonacci(n+1)+fibonacci(n-1);

%o a(n) = L(n)*L(n+1) % L(n+2); \\ _Michel Marcus_, Jan 24 2022

%Y Cf. A000032, A333599.

%K nonn,easy

%O 0,1

%A _J. M. Bergot_ and _Robert Israel_, Jan 23 2022