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 1-3 of 3 results.

A178428 5 followed by the generalized Fermat numbers 6^(2^n)+1 (A078303).

Original entry on oeis.org

5, 7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097
Offset: 0

Views

Author

Roger L. Bagula, May 27 2010

Keywords

Comments

If a(0) = 3, the recursion formula gives the Fermat numbers (A000215).
With a(0) = 3 instead of 5; a(n) = 2 + product_{i=0..n-1} a(i), n >= 1.
The recurrence equation for generalized Fermat numbers F_n(a) = a^(2^n)+1,
a >= 2, n >= 0, is F_{n}(a) = (F_{n-1}(a)-1)^2 + 1. - Daniel Forgues, Jun 22 2011

Crossrefs

Cf. A000215, A178427, A178426.

Programs

  • Mathematica
    a[0] := 5;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] + 2;
Table[a[n], {n, 0, 10}]

Formula

a(0) = 5; a(n) = 2 + product_{i=0..n-1} a(i), n >= 1.
From Daniel Forgues, Jun 22 2011: (Start)
The motivation for this sequence comes from the recurrence for generalized Fermat numbers 6^(2^n)+1 (A078303)
a(n) = 5*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 5*(empty product, i.e., 1)+ 2 = 7 = a(0). This implies that the terms are pairwise coprime. (End)

Extensions

Definition simplified by the Assoc. Eds. of the OEIS - May 28 2010
Edited by Daniel Forgues, Jun 22 2011

A178426 11 followed by the Fermat numbers A152585.

Original entry on oeis.org

11, 13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137
Offset: 0

Views

Author

Roger L. Bagula, May 27 2010

Keywords

Comments

If a(0)=3, the recursion formula gives A000215.

Crossrefs

Cf. A000215, A178427, A178428.

Programs

  • Mathematica
    a[0] := 11;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] + 2;
Table[a[n], {n, 0, 10}]

Formula

a(0)=11. a(n)= 2 + Product_{i=0..n-1} a(i).

Extensions

Definition simplified by the Assoc. Eds. of the OEIS - May 28 2010

A178497 5 followed by A145503.

Original entry on oeis.org

5, 3, 13, 193, 37633, 1416317953, 2005956546822746113, 4023861667741036022825635656102100993, 16191462721115671781777559070120513664958590125499158514329308740975788033
Offset: 0

Views

Author

Roger L. Bagula, May 28 2010

Keywords

Comments

The first 6 entries of the sequence are primes.
The Fermat numbers F(.)=A000215(.) obey equations with flipped signs of the +-2 if compared with the a(n) (see the formula section): F(n) = Prod{i=0,...,n-1} F(i)+2 and F(n) = F(n-1) *(F(n-1)-2)+2. - Vladimir Shevelev, Dec 08 2010

Crossrefs

Cf. A000215, A110407, A178428, A178427.

Programs

  • Mathematica
    Clear[a, n];
a[ -1] := 1; a[0] := 5;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] - 2;
Table[a[n], {n, 0, 10}]

Formula

a(n)= ( product_{i=0..n-1} a(i)) -2, n>=1.
a(n) = a(n-1)*( a(n-1)+2 )-2, n>=2. - Vladimir Shevelev, Dec 08 2010

Extensions

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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