cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A047946 a(n) = 5*F(n)^2 + 3*(-1)^n where F(n) are the Fibonacci numbers A000045.

Original entry on oeis.org

3, 2, 8, 17, 48, 122, 323, 842, 2208, 5777, 15128, 39602, 103683, 271442, 710648, 1860497, 4870848, 12752042, 33385283, 87403802, 228826128, 599074577, 1568397608, 4106118242, 10749957123, 28143753122, 73681302248, 192900153617, 505019158608, 1322157322202
Offset: 0

Views

Author

John W. Layman, May 21 1999

Keywords

Comments

Form the matrix A=[1,1,1;2,1,0;1,0,0]. a(n)=trace(A^n). - Paul Barry, Sep 22 2004
The set of prime divisors of elements of this sequence with the exception of 3 is the set of primes that do not divide odd Fibonacci numbers. - Tanya Khovanova, May 19 2008
If a(n) is prime then n is a power of 3 (Boase, 1998). The only values of k not exceeding 12 for which a(3^k) is prime are 0 and 1. - Amiram Eldar, Jun 19 2022

Links

Crossrefs

Cf. A000045, A000032, A005248.
Second row of array A028412.
Cf. A133247 (prime numbers p such that no odd Fibonacci number is divisible by p).

Programs

  • Mathematica
    Table[LucasL[n]^2 - (-1)^n, {n, 0, 30}] (* Amiram Eldar, Feb 02 2022 *)
  • PARI
    a(n)=5*fibonacci(n)^2+3*(-1)^n
  • Python
    from sympy import fibonacci
def A047946(n): return 5*fibonacci(n)**2+(-3 if n&1 else 3) # Chai Wah Wu, Jul 29 2022

Formula

a(n) = F(3n)/F(n), n>0.
a(n) = 2*a(n-1)+2*a(n-2)-a(n-3).
a(n) = 3a(n-1)-a(n-2)+5(-1)^n.
a(n) = A005248(n) + (-1)^n.
G.f.: ( 3-4*x-2*x^2 ) / ( (1+x)*(x^2-3*x+1) ).
for n>0 a(n) = A000045(3n)/A000045(n) - Benoit Cloitre, Aug 30 2003
For n>0, the linear recurrence for the sequence F(n*k)^2 has signature (a(n),a(n),-1) for n odd, and (a(n),-a(n), 1) for n even. For example, the linear recurrence for the sequence F(3*k)^2 has signature (17, 17, -1) (conjectured). - Greg Dresden, Aug 30 2021
a(n) = Lucas(n)^2 - (-1)^n. - Amiram Eldar, Feb 02 2022

Extensions

Entry improved by comments from Michael Somos.

A155916 Prime factors of odd Fibonacci numbers.

Original entry on oeis.org

3, 5, 7, 11, 13, 29, 37, 41, 43, 47, 59, 67, 71, 73, 89, 97, 101, 103, 113, 127, 131, 139, 149, 151, 157, 163, 179, 191, 193, 199, 223, 233, 239, 251, 263, 269, 277, 281, 283, 307, 311, 313, 331, 337, 347, 353, 359, 367, 373, 389, 397, 401, 419, 431, 433, 449
Offset: 1

Views

Author

Tanya Khovanova, Jan 30 2009

Keywords

Comments

Primes not in A133247.

Links

Crossrefs

Cf. A014437, A133247.

Programs

  • Maple
    filter:= proc(p) local a,b,i;
if not isprime(p) then return false fi;
a:= 0: b:= 1;
for i from 2 do
  a,b:= b, (a+b) mod p;
  if b = 0 then
    if i mod 3 <> 0 then return true
    elif a = 1 then return false
    fi
  fi
od:
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Nov 20 2016

A133246 Odd numbers n with property that no odd Fibonacci number is divisible by n.

Original entry on oeis.org

9, 17, 19, 23, 27, 31, 45, 51, 53, 57, 61, 63, 69, 79, 81, 83, 85, 93, 95, 99, 107, 109, 115, 117, 119, 133, 135, 137, 153, 155, 159, 161, 167, 171, 173, 181, 183, 187, 189, 197, 207, 209, 211, 217, 221, 225, 227, 229, 237, 241, 243, 247, 249, 253, 255, 257, 261
Offset: 1

Views

Author

Tanya Khovanova, Oct 14 2007, Oct 17 2007

Keywords

Comments

The Mathematica coding uses the fact that the Pisano period - the period with which a Fibonacci sequence (mod n) repeats itself is not more than 6n and the fact that the Fibonacci sequence starts with 0.

Links

Crossrefs

Cf. A133247.

Programs

  • Mathematica
    Transpose[ Select[Table[{k, Select[Table[{n, Mod[Fibonacci[n], k]}, {n, 6k + 1}], Mod[ #[[1]], 3] != 0 && #[[2]] == 0 &]}, {k, 1, 351, 2}], #[[2]] == {} &]][[1]]

A137189 Numbers n such that no odd Fibonacci is divisible by n and n is not a multiple of another number in this sequence.

Original entry on oeis.org

2, 9, 17, 19, 23, 31, 53, 61, 79, 83, 107, 109, 137, 167, 173, 181, 197, 211, 227, 229, 241, 257, 271, 293, 317, 349, 379, 383, 409, 421, 439, 443, 467, 499, 503, 541, 571, 587, 593, 601, 617, 631, 647, 653, 683, 691, 739, 751, 769, 773, 797, 811, 827, 829
Offset: 0

Views

Author

Tanya Khovanova, Mar 03 2008

Keywords

Comments

Obviously if no odd Fibonacci is divisible by n, then no odd Fibonacci is divisible by a multiple of n.
The only composite in this sequence is 9.

Examples

			The most famous is the statement that no odd Fibonacci is divisible by 17.
Trivially, no odd Fibonacci is divisible by 2.

Links

Crossrefs

Cf. A133247 Prime numbers p with property that no odd Fibonacci number is divisible by p. A133246 Odd numbers n with property that no odd Fibonacci number is divisible by n.
Showing 1-4 of 4 results.

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