A357553 a(n) = A000045(n)*A000045(n+1) mod A000032(n).

0, 0, 2, 2, 1, 7, 14, 12, 9, 46, 98, 80, 64, 313, 674, 546, 441, 2143, 4622, 3740, 3025, 14686, 31682, 25632, 20736, 100657, 217154, 175682, 142129, 689911, 1488398, 1204140, 974169, 4728718, 10201634, 8253296, 6677056, 32411113, 69923042, 56568930, 45765225, 222149071, 479259662, 387729212

Offset: 0

Author

J. M. Bergot and Robert Israel, Oct 02 2022

Keywords

• nonn

Comments

a(n) is the product of the n-th and (n+1)th Fibonacci numbers mod the n-th Lucas number.

Examples

			a(3) = A000045(3)*A000045(4) mod A000032(3) = 2*3 mod 4 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..4761

Crossrefs

Cf. A000032, A000045, A333599, A347861.

Programs

Maple

luc:= n -> combinat:-fibonacci(n+1) + combinat:-fibonacci(n-1):
f:= proc(n) local m;
  m:= n mod 4;
  if m = 0 then (luc(n)-2)/5
  elif m = 1 then (3*luc(n)+2)/5
  elif m = 2 then (4*luc(n)-2)/5
  else (2*luc(n)+2)/5
  fi
end proc:
f(1):= 0:
map(f, [$0..50]);

Mathematica

a[n_] := Mod[Fibonacci[n] * Fibonacci[n + 1], LucasL[n]]; Array[a, 50, 0] (* Amiram Eldar, Oct 03 2022 *)

Formula

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).

For n == 0 (mod 4), a(n) = (A000032(n) - 2)/5.

For n == 1 (mod 4) and n > 1, a(n) = (3*A000032(n) + 2)/5.

For n == 2 (mod 4), a(n) = (4*A000032(n) - 2)/5.

For n == 3 (mod 4), a(n) = (2*A000032(n) + 2)/5.