A105183 a(n) = 1 + a(n-1)*(a(n-1) + 1), with a(0)=2.

2, 7, 57, 3307, 10939557, 119673918295807, 14321846720271609085072077057, 205115293478954645768397227034180943592279329877217858307, 42072283618957694230389567430137958296609066493047345973782287300661413651741392431587718724877522268597146764557

Offset

0,1

Comments

For n > 1, a(n) has digital root 3 or 4 depending on whether n is odd or even.

The 5-adic valuation of a(n)^2 + 1 is n+1. - William Hu, Dec 01 2023

References

T. Koshy, "Intriguing Properties Of Three Related Number Sequences", in Journal of Recreational Mathematics, vol. 32(3) 210-3 2003-4 Baywood NY.

Links

Table of n, a(n) for n=0..8.

Formula

From Gerald McGarvey, Dec 12 2007: (Start)

For n > 0, a(n) = Sum_{k=0..n-1} a(k)^2 + n + 2.

Conjecture: a(n) is asymptotic to d - 1/2 -(5/2^3)/d -(65/2^7)/d^3 -(650/2^11)/d^5 -(19045/2^15)/d^7 -(274950/2^19)/d^9 -(6979050/2^23)/d^11 -(130292500/2^27)/d^13 ... where d = c^(2^n) and c is a constant = 1.288203192684485177845610784851700404829443712770079185959554466777577486352420255603915828361833141546.... (End)

Maple

a[0]:=2: for n from 1 to 8 do a[n]:=1+a[n-1]*(a[n-1]+1) od: seq(a[n], n=0..8); # Emeric Deutsch, Jun 13 2005

Mathematica

NestList[1+#(#+1)&, 2, 10] (* Harvey P. Dale, Mar 30 2016 *)

Crossrefs

Cf. A005267.

Sequence in context: A178769 A121079 A270395 * A269994 A023364 A120952

Adjacent sequences: A105180 A105181 A105182 * A105184 A105185 A105186

Keyword

nonn

Author

Lekraj Beedassy, Apr 11 2005

Extensions

More terms from Emeric Deutsch, Jun 13 2005

Status

approved