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%I #17 Oct 12 2022 08:57:56
%S 0,0,2,2,1,7,14,12,9,46,98,80,64,313,674,546,441,2143,4622,3740,3025,
%T 14686,31682,25632,20736,100657,217154,175682,142129,689911,1488398,
%U 1204140,974169,4728718,10201634,8253296,6677056,32411113,69923042,56568930,45765225,222149071,479259662,387729212
%N a(n) = A000045(n)*A000045(n+1) mod A000032(n).
%C a(n) is the product of the n-th and (n+1)th Fibonacci numbers mod the n-th Lucas number.
%H Robert Israel, <a href="/A357553/b357553.txt">Table of n, a(n) for n = 0..4761</a>
%F G.f. (2 + 2*x + 3*x^2 + 7*x^3 + 3*x^4 + 2*x^5 + x^6)*x^2/(1 + x^2 - x^3 - 5*x^4 - 3*x^5 - 2*x^6 - x^7).
%F For n == 0 (mod 4), a(n) = (A000032(n) - 2)/5.
%F For n == 1 (mod 4) and n > 1, a(n) = (3*A000032(n) + 2)/5.
%F For n == 2 (mod 4), a(n) = (4*A000032(n) - 2)/5.
%F For n == 3 (mod 4), a(n) = (2*A000032(n) + 2)/5.
%e a(3) = A000045(3)*A000045(4) mod A000032(3) = 2*3 mod 4 = 2.
%p luc:= n -> combinat:-fibonacci(n+1) + combinat:-fibonacci(n-1):
%p f:= proc(n) local m;
%p m:= n mod 4;
%p if m = 0 then (luc(n)-2)/5
%p elif m = 1 then (3*luc(n)+2)/5
%p elif m = 2 then (4*luc(n)-2)/5
%p else (2*luc(n)+2)/5
%p fi
%p end proc:
%p f(1):= 0:
%p map(f, [$0..50]);
%t a[n_] := Mod[Fibonacci[n] * Fibonacci[n + 1], LucasL[n]]; Array[a, 50, 0] (* _Amiram Eldar_, Oct 03 2022 *)
%Y Cf. A000032, A000045, A333599, A347861.
%K nonn
%O 0,3
%A _J. M. Bergot_ and _Robert Israel_, Oct 02 2022
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