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

0,3

Comments

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

Links

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

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.

Example

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

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 *)

Crossrefs

Cf. A000032, A000045, A333599, A347861.

Sequence in context: A360377 A100632 A225925 * A246745 A111540 A360410

Adjacent sequences: A357550 A357551 A357552 * A357554 A357555 A357556

Keyword

nonn

Author

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

Status

approved