OFFSET
1,1
COMMENTS
The sets begin {1,2}, {3,4,5}, {6,7,8,...,17}, ...
Starting with {1}, one would get {1}, {2}, {3,4}, {5,6,7, 8,9,10, 11} ... with sums (1,2,7,56, 2212 ...) = A002658. - M. F. Hasler, Jan 21 2015
FORMULA
a(n)= a(n-1) *( 1 +2*[a(1)+a(2)+...+a(n-2)] +a(n-1) )/2. - Corrected by R. J. Mathar, Jan 22 2015
a(n) = a(n-1)*(2*a(n-1) + a(n-2)*a(n-1) + a(n-2)^2)/(2*a(n-2)). - David W. Wilson, Jan 22 2015
a(n+1) = a(n)*(2*A067339(n)-a(n)+1)/2. - M. F. Hasler, Jan 23 2015
a(n) ~ 2 * c^(2^n), where c = 1.312718001584962838462131787518361199185077166417566246117... . - Vaclav Kotesovec, Dec 09 2015
MAPLE
# Return [start, number, sum] of n-th group
A067338aux := proc(n)
local StrNumSu, Strt, Num, Su ;
option remember;
if n = 1 then
return [1, 2, 3] ;
else
strNumSu := procname(n-1) ;
Strt := strNumSu[1]+strNumSu[2] ;
Num := strNumSu[3] ;
Su := Num*(Num+2*Strt-1)/2 ;
return [Strt, Num, Su] ;
end if;
end proc:
A067338 := proc(n)
A067338aux(n)[2] ;
end proc:
seq(A067338(n), n=1..10) ; # R. J. Mathar, Jan 21 2015
MATHEMATICA
RecurrenceTable[{a[n] == a[n-1]*(2*a[n-1] + a[n-2]*a[n-1] + a[n-2]^2)/(2*a[n-2]), a[1]==2, a[2]==3}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 09 2015 *)
PROG
(PARI) print1(a=n=2); for(i=2, 9, print1(", "n=n*(a+a-n+1)/2); a+=n) \\ M. F. Hasler, Jan 21 2015
(Magma) I:=[2, 3]; [n le 2 select I[n] else Self(n-1)*(2*Self(n-1) + Self(n-2)*Self(n-1) + Self(n-2)^2)/(2*Self(n-2)): n in [1..10]]; // Vincenzo Librandi, Jan 23 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Floor van Lamoen, Jan 16 2002
EXTENSIONS
Corrected and extended by Harvey P. Dale and M. F. Hasler, Jan 21 2015
STATUS
approved