A348592 a(n) = F(n)*F(n+1) mod L(n+2) where F=A000045 is the Fibonacci numbers and L = A000032 is the Lucas numbers.

0, 1, 2, 6, 15, 11, 10, 45, 99, 79, 65, 312, 675, 545, 442, 2142, 4623, 3739, 3026, 14685, 31683, 25631, 20737, 100656, 217155, 175681, 142130, 689910, 1488399, 1204139, 974170, 4728717, 10201635, 8253295, 6677057, 32411112, 69923043, 56568929, 45765226, 222149070, 479259663, 387729211, 313679522

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..4759

Index entries for linear recurrences with constant coefficients, signature (0,-1,1,5,3,2,1).

Formula

For n >= 1, a(n) = (A070352(n+2)*A000032(n+2) + 3*(-1)^n)/5.

a(n) + 2*a(n + 1) + 3*a(n + 2) + 5*a(n + 3) + a(n + 4) - a(n + 5) - a(n + 7) = 0 for n >= 1.

G.f.: -3 + 3/(5*(1+x)) + (3+x)/(2*(1-x-x^2)) + (9-4*x+6*x^2-x^3)/(10*(1+3*x^2+x^4)).

Example

a(5) = F(5)*F(6) mod L(7) = 5*8 mod 29 = 11.

Maple

F:= combinat:-fibonacci:
L:= n -> F(n-1)+F(n+1):
seq(F(n)*F(n+1) mod L(n+2), n=0..20);

Mathematica

a[n_] := Mod[Fibonacci[n] * Fibonacci[n + 1], LucasL[n + 2]]; Array[a, 50, 0] (* Amiram Eldar, Jan 26 2022 *)

Crossrefs

Cf. A000032, A000045, A070352, A333599, A347861, A348591.

Sequence in context: A297574 A349984 A215081 * A337737 A119416 A134891

Adjacent sequences: A348589 A348590 A348591 * A348593 A348594 A348595

Keyword

nonn,easy

Author

J. M. Bergot and Robert Israel, Jan 25 2022

Status

approved