A172028 a(1) = 2; for n > 1, a(n) = smallest k such that a(n-1)^3+k is a cube.

2, 19, 1141, 3909067, 45842426158669, 6304584108339196948770030691, 119243342337369441148530917575983600568317619049555904517

Offset

1,1

Comments

a(8) has 113 decimal digits.

From a(2) onward a subsequence of A003215 (centered hexagonal numbers: 3n(n+1)+1, also first differences of A000578). - Klaus Brockhaus, Mar 20 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..11

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Formula

a(n) = 1 + 3*a(n-1)*(a(n-1) + 1). - R. J. Mathar, Jan 25 2010

a(n) ~ k^(2^n) * 3^n with k = 2.7658.... - Charles R Greathouse IV, Dec 29 2011

The constant k = 2.765824704990104629134798316783956... . - Vaclav Kotesovec, Jan 19 2015

Example

n = 2: for k = 19, a(1)^3+k = 2^3+19 = 27 = 3^3 is a cube; 19 is the smallest such k, therefore a(2) = 19.
n = 4: for k = 3909067, a(3)^3+k = 1141^3+3909067 = 1489355288 = 1142^3 is a cube; 3909067 is the smallest such k, therefore a(4) = 3909067.

Maple

A172028 := proc(n) option remember; if n <=2 then op(n, [2, 19]) : else 1+3*procname(n-1)*(procname(n-1)+1); end if; end: seq(A172028(n), n=1..8) ; # R. J. Mathar, Jan 25 2010

Mathematica

Join[{a = 2}, Table[a = (a + 1)^3 - a^3, {n, 8}]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
RecurrenceTable[{a[n]==1+3*a[n-1]*(1+a[n-1]), a[1]==2}, a, {n, 1, 10}] (* Vaclav Kotesovec, Jan 19 2015 *)

Prog

(Magma) /* inefficient, uses definition */ a:=2; S:=[a]; for n in [2..4] do k:=0; flag:= true; while flag do k+:=1; if IsPower(a^3+k, 3) then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
/* uses formula from R. J. Mathar */ [ n eq 1 select 2 else 1+3*Self(n-1)*(Self(n-1)+1): n in [1..8] ]; // Klaus Brockhaus, Mar 16 2010

Crossrefs

Cf. A172027.

Sequence in context: A013107 A369946 A350933 * A365050 A024229 A094663

Adjacent sequences: A172025 A172026 A172027 * A172029 A172030 A172031

Keyword

nonn

Author

Vincenzo Librandi, Jan 23 2010

Extensions

Edited, a(4), a(5), a(6) corrected, a(7) added by Klaus Brockhaus, Mar 16 2010

Status

approved