%I #27 Sep 08 2022 08:45:50
%S 2,19,1141,3909067,45842426158669,6304584108339196948770030691,
%T 119243342337369441148530917575983600568317619049555904517
%N a(1) = 2; for n > 1, a(n) = smallest k such that a(n-1)^3+k is a cube.
%C a(8) has 113 decimal digits.
%C 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
%H Vincenzo Librandi, <a href="/A172028/b172028.txt">Table of n, a(n) for n = 1..11</a>
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F a(n) = 1 + 3*a(n-1)*(a(n-1) + 1). - _R. J. Mathar_, Jan 25 2010
%F a(n) ~ k^(2^n) * 3^n with k = 2.7658.... - _Charles R Greathouse IV_, Dec 29 2011
%F The constant k = 2.765824704990104629134798316783956... . - _Vaclav Kotesovec_, Jan 19 2015
%e 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.
%e 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.
%p 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
%t Join[{a = 2}, Table[a = (a + 1)^3 - a^3, {n, 8}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 24 2012 *)
%t RecurrenceTable[{a[n]==1+3*a[n-1]*(1+a[n-1]),a[1]==2},a,{n,1,10}] (* _Vaclav Kotesovec_, Jan 19 2015 *)
%o (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;
%o /* 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
%Y Cf. A172027.
%K nonn
%O 1,1
%A _Vincenzo Librandi_, Jan 23 2010
%E Edited, a(4), a(5), a(6) corrected, a(7) added by _Klaus Brockhaus_, Mar 16 2010