A296397 a(n) = a(n-1) * a(n-2) + a(n-3) * Product_{k=0..n-4} a(k)^2, with a(0) = a(1) = 1, a(2) = 2.

1, 1, 2, 3, 7, 23, 173, 4231, 772535, 3430031573, 2767984611331999, 9880508763685677890784167, 28372546978138838124644984908123272195533, 290052121708444744262218759616469916140851065875997330620050069911

Offset

0,3

Comments

The recurrence for b(n) is similar to Fibonacci except for the reciprocal.

An infinite coprime sequence defined by recursion. - Michael Somos, Dec 14 2017

Links

Table of n, a(n) for n=0..13.

Michael Penn, 2 viewer suggested recursive sequences., YouTube video, 2021.

Larry Powell, Any insight in the half reciprocal Fibonacci sequence? Math StackExchange, Dec 11 2017.

Formula

a(n) = a(n-1) * a(n-2) + a(n-1) * a(n-3) * a(n-4) - a(n-2) * a(n-3)^2 * a(n-4) for all n>=4.

a(n) = numerator of b(n) where b(0) = b(1) = 1, b(n) = b(n-1) + 1/b(n-2).

Mathematica

a[ n_] := Which[ n < 1, Boole[n == 0], n < 4, n, True, a[n - 1] a[n - 2] + a[n - 3] Product[ a[k], {k, 0, n - 4}]^2];
Numerator@ RecurrenceTable[{a[n] == a[n - 1] + 1/a[n - 2], a[0] == a[1] == 1}, a, {n, 0, 13}] (* Robert G. Wilson v, Dec 11 2017 *)

Prog

(PARI) {a(n) = if( n<1, n==0, n<4, n, a(n-1) * a(n-2) + a(n-3) * prod(k=0, n-4, a(k))^2)};

Crossrefs

Sequence in context: A111235 A066356 A006892 * A102710 A048824 A355015

Adjacent sequences: A296394 A296395 A296396 * A296398 A296399 A296400

Keyword

nonn

Author

Michael Somos, Dec 11 2017

Status

approved