A143766 a(n+1) = a(n)^2 + 3*n*a(n) + n^2, a(1) = 1.

1, 5, 59, 4021, 16216709, 262981894041341, 69159476593575838635509822455, 4783033202697364284917104840982811414253511628131328498629, 22877406618105405861781317490149379589769149890660405723416585348109182559037843469373513563751798569651299138846801

Offset

1,2

Comments

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:

f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),

f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n), f(n,+3)=a(n).

Links

Table of n, a(n) for n=1..9.

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From Reinhard Zumkeller, Sep 11 2008]

Dario A. Alpern, Factorization using the Elliptic Curve Method [From Reinhard Zumkeller, Sep 11 2008]

Formula

a(n) ~ c^(2^n), where c = 1.68000796750332615134775696497253700657744224375254906378714756508286... . - Vaclav Kotesovec, Dec 18 2014

Example

Contribution from Reinhard Zumkeller, Sep 11 2008: (Start)
a(4)=A000040(556);
a(5)=19*199*4289;
a(6)=3686299*71340359;
a(7)=5*89*23581*36190079671*182112572569;
A055642(a(8))=58; A001221(a(8))=A001222(a(8))=3;
A055642(A020639(a(8)))=4, A020639(a(8))=2459;
A055642(A006530(a(8)))=43, A006530(a(8))=1145781805709434583439407716589323093429591;
A055642(a(9))=116; A001221(a(9))=A001222(a(9))=3;
A055642(A020639(a(9)))=5, A020639(a(9))=52291;
A055642(A087039(a(9)))=39, A087039(a(9))=823717865733493312451872329574156137131;
A055642(A006530(a(9)))=72, A006530(a(9))=531130643259166452223939782963931943654770628199012648274446497807560081;
factorizations made with Dario Alpern's ECM applet. (End)

Mathematica

RecurrenceTable[{a[n+1] == a[n]^2 + 3*n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

Crossrefs

Sequence in context: A326573 A324240 A120608 * A132549 A091457 A289724

Adjacent sequences: A143763 A143764 A143765 * A143767 A143768 A143769

Keyword

nonn

Author

Reinhard Zumkeller, Sep 01 2008

Status

approved