A022462 -id:A022462 - cp's OEIS frontend

A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

0, 1, 2, 1, 7, 1, 20, 1, 54, 1, 143, 1, 376, 1, 986, 1, 2583, 1, 6764, 1, 17710, 1, 46367, 1, 121392, 1, 317810, 1, 832039, 1, 2178308, 1, 5702886, 1, 14930351, 1, 39088168, 1, 102334154, 1, 267914295, 1, 701408732, 1, 1836311902, 1, 4807526975, 1, 12586269024

Offset: 0

Adnan Baysal, Mar 28 2020

			a(0) = 0*1 mod 1 = 0;
a(1) = 1*1 mod 2 = 1;
a(2) = 1*2 mod 3 = 2;
a(3) = 2*3 mod 5 = 1;
a(4) = 3*5 mod 8 = 7.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, d'Ocagne's Identity.
Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-1,-1).

Equals A035508 interleaved with A000012.

Cf. A182126, A000045, A022461, A072565, A022462.

With[{f = Fibonacci}, Table[Mod[f[n] * f[n+1], f[n+2]], {n, 0, 50}]] (* Amiram Eldar, Mar 28 2020 *)

a(n) = if (n % 2, 1, fibonacci(n+2) - 1); \\ Michel Marcus, Mar 29 2020

concat(0, Vec(x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)) + O(x^45))) \\ Colin Barker, Mar 29 2020

def a(n):
    f1 = 0
    f2 = 1
    for i in range(n):
        f = f1 + f2
        f1 = f2
        f2 = f
    return (f1 * f2) % (f1 + f2)

a(2n+1) = 1, and a(2n) = F(2n+2) - 1, and lim(a(2n+2)/a(2n)) = phi^2 by d'Ocagne's identity.
a(n) = F(n) * F(n+1) mod (F(n) + F(n+1)) since F(n+2) := F(n+1) + F(n).
From Colin Barker, Mar 28 2020: (Start)
G.f.: x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)).
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n>4.
(End)

A073271 a(n) = floor( prime(n)*prime(n+2) / prime(n+1) ).

Original entry on oeis.org

3, 4, 7, 8, 14, 14, 20, 23, 24, 34, 34, 38, 44, 48, 52, 54, 64, 64, 68, 76, 76, 84, 90, 92, 98, 104, 104, 110, 122, 116, 132, 132, 146, 140, 154, 156, 160, 168, 172, 174, 188, 182, 194, 194, 208, 210, 214, 224, 230, 234, 234, 248, 246, 256, 262, 264, 274, 274

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

A000040(n) < a(n) < A000040(n+2);
Conjecture: a(n) is even except for a(1)=3, a(3)=7 and a(8)=23 (A073272(n)=0).
Conjecture: when a(n) = a(n+1) then A001223(n+1) = A001223(n) + A001223(n+2). - Gionata Neri, May 31 2015

			A000040(10)*A000040(12)/A000040(11) = 29*37/31 = 1073/31 = (34*31+19)/31, therefore a(10)=34; A073272(10) = A000040(11)-a(10) = 31-34 = -3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A022462, A073272, A073273.

[Floor(NthPrime(n)*NthPrime(n+2) / NthPrime(n+1)): n in [1..60]]; // Vincenzo Librandi, May 31 2015

Table[Floor[Prime[n] Prime[n + 2] / Prime[n + 1]], {n, 60}] (* Vincenzo Librandi, May 31 2015 *)
Floor[(#[[1]]#[[3]])/#[[2]]]&/@Partition[Prime[Range[60]],3,1] (* Harvey P. Dale, Jun 07 2021 *)

vector(100, n, (prime(n)*prime(n+2))\prime(n+1)) \\ Michel Marcus, May 31 2015

A182126 a(n) = prime(n)*prime(n+1) mod prime(n+2).

Original entry on oeis.org

1, 1, 2, 12, 7, 12, 1, 2, 16, 11, 40, 12, 24, 7, 13, 16, 48, 40, 12, 48, 40, 60, 15, 48, 12, 24, 12, 24, 125, 72, 60, 16, 120, 24, 48, 72, 40, 60, 72, 16, 120, 24, 24, 12, 168, 65, 64, 12, 24, 60, 16, 120, 96, 72, 72, 16, 48, 40, 12, 120, 29, 72, 12, 24, 252

Offset: 1

Alex Ratushnyak, Apr 13 2012

Conjecture: for x>10^9, the most frequent value in a(n), n=0...x, has form 120*k.
Let b = prime(n+2) - prime(n) and c = prime(n+2) - prime(n+1). Conjecture: for n > 61, a(n) = b*c. This holds up to n = 9 * 10^16. - Charles R Greathouse IV, May 11 2012
With b and c as above, a(n) = b*c if and only if b*c < prime(n+2). Cramér's conjecture implies this is true for all sufficiently large n. - Robert Israel, Jun 19 2017

			(2*3) mod 5 = 1, (3*5) mod 7 = 1, (5*7) mod 11 = 2, (7*11) mod 13 = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A022461, A022462.

a182126 n = a182126_list !! (n-1)
a182126_list = zipWith3 (\p p' p'' -> mod (p * p') p'')
                  a000040_list (tail a000040_list) (drop 2 a000040_list)
-- Reinhard Zumkeller, Apr 23 2012

[NthPrime(n)*NthPrime(n+1) mod NthPrime(n+2): n in [1..70]]; // Vincenzo Librandi, Jun 20 2017

P:= [seq(ithprime(i),i=1..102)]:
seq(P[i]*P[i+1] mod P[i+2], i=1..100); # Robert Israel, Jun 19 2017

Mod[#[[1]]#[[2]],#[[3]]]&/@Partition[Prime[Range[70]],3,1] (* Harvey P. Dale, Sep 30 2015 *)

p=2;q=3;forprime(r=5,1e3,print1(p*q%r", ");p=q;q=r) \\ Charles R Greathouse IV, May 11 2012

cp's OEIS Frontend

A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

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A073271 a(n) = floor( prime(n)*prime(n+2) / prime(n+1) ).

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A182126 a(n) = prime(n)*prime(n+1) mod prime(n+2).

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Haskell

Magma

Maple

Mathematica

PARI